Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Article Dynamics of Rössler Prototype-4 System: Analytical and Numerical Investigation

Svetoslav G. Nikolov 1,2,* and Vassil M. Vassilev 1

1 Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., bl. 4, 1113 Soﬁa, Bulgaria; [email protected] 2 Department of Mechanics, University of Transport, Geo Milev Str., 158, 1574 Soﬁa, Bulgaria * Correspondence: [email protected]; Tel.: +359-2-979-6411

Abstract: In this paper, the dynamics of a 3D autonomous dissipative nonlinear system of ODEs- Rössler prototype-4 system, was investigated. Using Lyapunov-Andronov theory, we obtain a new analytical formula for the ﬁrst Lyapunov’s (focal) value at the boundary of stability of the corresponding equilibrium state. On the other hand, the global analysis reveals that the system may exhibit the phenomena of Shilnikov chaos. Further, it is shown via analytical calculations that the considered system can be presented in the form of a linear oscillator with one nonlinear automatic regulator. Finally, it is found that for some new combinations of parameters, the system demonstrates chaotic behavior and transition from chaos to regular behavior is realized through inverse period-doubling bifurcations.

Keywords: analysis; chaos; Rössler prototype-4 system; analytical and numerical investigation

1. Introduction Citation: Nikolov, S.G.; Vassilev, V.M. In the last thirty years, many authors have been aroused by the search for the mathe- Dynamics of Rössler Prototype-4 matically simplest systems of various species that can exhibit chaos [1–7]. A good example System: Analytical and Numerical is the book of J. Sprott [4] in which he discovered: (1) some new systems that are simpler Investigation. Mathematics 2021, 9, than those previously know; (2) these new systems are otherwise more “elegant” on the 352. https://doi.org/10.3390/math part of the number of system parameters-their values, special symmetry and economy 9040352 of notation. The theory of dynamical systems which describes the qualitative changes in phase Academic Editor: Borislav Stoyanov spaces at a variation of one or several parameters is called bifurcation theory [7–11]. Gen- Received: 5 January 2021 erally, all bifurcations can be separated (classed) into two main kinds: local and global. Accepted: 1 February 2021 Published: 10 February 2021 Although it is difficult to make a precise distinction between them, the dynamics of local bifurcations is determined by information contained in the terms of Taylor series or `е Publisher’s Note: MDPI stays neutral Poincar map at a point, and the dynamics of global bifurcations requires information about with regard to jurisdictional claims in the vector field along an entire (non-periodic) orbit [8–10]. With respect to dynamical published maps and institutional affil- systems theory [9,10,12], the Hopf bifurcation theorem [13], and some other elements of the iations. local bifurcation theory (e.g., the first Lyapunov value—L1(λ0)) are basic analytical instru- ments to investigate the transition between different dynamical states in a neighborhood of equilibrium states (fixed points). The (in) stability of these transitions may have essential consequences on the dynamics of the system. Practically, the complete investigation of complex phenomena in a dynamical system is impossible without a thorough study of Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. both bifurcations-local and global. This article is an open access article An important element towards the understanding of the global behavior of a dy- distributed under the terms and namical system of nonlinear differential equations is the analysis of the existence of ho- conditions of the Creative Commons moclinic/heteroclinic trajectories (cycles) [14–17]. Note that a classical homoclinic cycle Attribution (CC BY) license (https:// Γ0 is a loop that consists of a saddle equilibrium state, i.e., this as a trajectory which is creativecommons.org/licenses/by/ bi-asymptotical to a saddle periodic orbit as time t → ±∞ . According to Peixoto’s theo- 4.0/). rem [18,19], homoclinic bifurcations are structurally unstable, i.e., they can be destroyed by

Mathematics 2021, 9, 352. https://doi.org/10.3390/math9040352 https://www.mdpi.com/journal/mathematics Mathematics 2021, 9, 352 2 of 17

small perturbations. Therefore, they are more difficult to identify than a local bifurcation, because knowledge of the global properties of the phase space trajectories is required. Ac- cording to [11,20], the bifurcations in the dynamical systems can be divided into three types: (1) bifurcations not originating from the Morse-Smale class of systems; (2) bifurcations in a class of systems with non-trivial hyperbolicity; and (3) bifurcations associated with the transition from Morse-Smale to systems with non-trivial hyperbolicity. The last bifurcations are of particular interest because they can explain the mechanisms of emergence of chaotic properties in deterministic dynamical systems. It is well-known that chaos cannot occur in dissipative two-dimensional systems (one degree of freedom) of ordinary differential equations (ODEs). Chaos requires systems with at least one and a half degrees of freedom. Such systems have so-called strange attractors (the trajectory winds around forever, never repeating) with non-integer dimension. In a three-dimensional continuous dissipative dynamical systems, the only possible spectra, and the attractors, are as follows: a strange attractor, a two-torus, a limit cycle, a fixed point [21–23]. Mathematical representation of a spatial order and chaos are saddle equilibria, saddle periodic movements or complex saddle invariant set. According to [24], around a saddle-focus equilibrium a systematic characterization of homoclinicity can be provided. It is well-known that, if the Shilnikov condition is satisfied, i.e., the saddle-focus index χ δ = |Re( 2/χ1)| < 1 (where χ1 (or χs) and χ2 (or χu) are the leading stable and unstable eigenvalues (manifolds)), then an infinite number of non-periodic trajectories coexist in the vicinity of a homoclinic trajectory (orbit) Γ0 bi-asymptotic to the saddle-focus. For more details see [19,24,25]. In other words, if exactly one of the leading eigenvalues is complex (i.e., χs ∈ C and χu ∈ R), then the homoclinic orbit is called unifocal and according to the Shilnikov condition close to it: (1) there exist horseshoes in every neighborhood of Γ0-when δ > 1; and (2) there is a neighborhood of Γ0 which contains no periodic orbits- when δ < 1. Here we note that Shilnikov condition δ 6= 1 is also called non-resonance condition [15,16,26]. If we look at further conditions on the eigenvalues we can determine a second genericity requirement: δ 6= 2. If 1 < δ < 2, then the linear flow at the equilibrium state(s) is contracting and the horseshoes are attracting, while for δ > 2 the linear flow at the origin is expanding [16]. The generic homoclinic orbit with complex principal eigenvalues (bifocal homoclinic orbit) has been studied by numerous researchers [25,27–29]. In a theorem they prove that horseshoes exist in every neighborhood of a generic bifocal homoclinic orbit. For a long period of time the Lorenz and Rössler systems [30,31] were regarded as the simplest chaotic autonomous dissipative systems of ODEs. In this paper we consider the so-called Rössler prototype-4 system given by

. x = −y − z, . y = x, (1) . z = −bz + ay − y2,

where (x, y, z) ∈ R3 are the state variables [1]. Originally, Rössler in [1] has proposed a few abstract three-component chemical models (systems) which have chaotic solutions. Latter, Olsen and Degn [32,33] showed that these systems (including system (1)) are an example of chaos in experimental (real) chemical (enzyme) reaction systems. It is seen that system (1) has only two parameters, six terms and a single quadratic nonlinearity. To the best of our knowledge, the system was only numerically studied in [1,3,4,34]. It was shown that for a = b = 0.5 the system (1) has chaotic behavior (see for more details Figure A3 in AppendixC), and for a ∈ (0, 1], b ∈ (0, 1] has different dynamics-unbounded, periodic and chaotic solutions. In recent years, some interesting results about stability of periodic orbit and possible modiﬁcations of Rössler prototype-4 system (1) have emerged. In Garcia et al. [35] existence and stability of periodic orbits of system (1) are proved. The work described in Sprott and Linz [36] is another example where for some values of the parameters the dynamics of the system (1) is quasiperiodic and a trajectory lies on an invariant torus. In [37,38] the authors: (1) studied the condition and type of Hopf Mathematics 2021, 9, 352 3 of 17

bifurcation in system (1) based on normal form; (2) investigated the amplitude control of limit cycle; (3) modiﬁed system (1) based on Chua’s diode nonlinearity; and (4) studied multistability and multiscroll generation of system (1). However, a detailed analysis of the transitory phenomena leading to irregular (chaotic) dynamics of system (1) is missing, which prompted our interest. In this paper, using analytical and numerical tools, we investigate the dynamical behavior of system (1). The plan is as follows: in Sections2 and3 we present analytical and numerical results concerning the system (1) for different values of bifurcation (control) parameters a and b. In Section4 we summarize our results.

2. Analytical Investigation 2.1. Local Analysis and First Lyapunov Value-L1 In this section, we are going to consider the system (1) which presents an autonomous dynamical model. Clearly, the equilibrium (steady state) points of system (1) are: O1(0, 0, 0) b b and O2 0, 1 + a , −1 − a . The divergence of the ﬂow (1) is:

. . . ∂x ∂y ∂z D (t) = + + = D (t ) × e−bt = −b, (2) 3 ∂x ∂y ∂z 3 0

where D3(t0) is a volume element. The system (1) is dissipative and has an attractor, when D3(t) < 0, i.e., b > 0. The system (1) can be presented in the form of a (non)-linear oscillator with one automatic regulator. In general form we have

.. ∂V . y = − , z = −ε[z − g(y)], (3) ∂y

where ε is a small parameter, g(y) is a nonlinear polynomial function and V is the potential 4 2 i α0y y in the form V = 2 − ∑ (αi − βiz) i [39]. For system (1) we have: α0 = α1 = β2 = 0, i=1 α2 = −1 and β1 = 1. Hence, for the system (1) we can write

.. ∂V y = − ∂y = −y − z, . (4) z = −b[z − kg(y)] = −bz + ay − y2,

2 a y where k = /b, and the potential of the system is V = 2 + yz. The energy of the system (1) . 2 y is E = V + 2 . Here we note that Lorenz system also can be presented in the form (3) as 2 α0 = 1, α1 = α2 = β1 = 0, β2 = 1 and g(y) = y [39]. To determine analytically ﬁrst Lyapunov value (L1) and Routh-Hurwitz conditions for stability of ﬁxed points, we must accomplish some transformations of system (1). Hence, we obtain ...... y = −by − y − (a + b)y + ay2. (5) Let us denote w = y − y0. (6) After substitution of (6) into (5), Equation (5) takes the form ...... w = −bw − w + cw + aw2, (7)

b where c = a(2y0 − 1) − b and y0 is the equilibrium states O1(y = 0) or O2 y = 1 + a . On the other hand, let .. . w = y1, w = y2, w = y3. (8) Mathematics 2021, 9, 352 4 of 17

Substituting (8) in (7), we obtain a system of three ﬁrst-order differential equations in the form . 2 y = −by1 − y2 + cy3 + ay , . 1 3 y = y1, (9) . 2 y3 = y2.

The Routh-Hurwitz conditions for stability of O1 and O2 can be written in the form [12]: p = b > 0, q = 1 > 0, (10) r = −c = −a(2y0 − 1) + b > 0, R = pq − r = b + c = a(2y0 − 1) > 0. The notations p, q, r and R are taken from [12]. The characteristic equation of the system (9) (which is equivalent to system (1)) has the form

χ3 + pχ2 + qχ + r = 0. (11)

According to [9,12], the boundaries of the stability region for three-dimensional systems, are two surfaces: Ψ1(R = 0, p > 0, q > 0) and Ψ2(r = 0, p > 0, q > 0). On the first surface R = 0, the characteristic equation (11) has a pair of purely imaginary roots as in this case an Andronov-Hopf (AH) bifurcation takes place, and at least one zero root on the surface r = 0. It is well-known that Andronov-Hopf bifurcation can be: (i) supercritical (soft loss of stability) and (ii) subcritical (hard loss of stability) [9,12,13,23,40]. In the bifurcation points (where R = 0), a positive first Lyapunov value represents a subcritical (hard or irreversible) bifurcation and determines that the system possesses unstable solutions but may fold back and exhibit unstable periodic solutions (oscillations) (unstable limit cycle) coexisting with stable steady state. In this case the boundary of stability is called “dangerous”. Inversely, a negative value for L1 predicts a supercritical (soft or reversible) Andronov-Hopf bifurcation. Thus, the loss of stability takes place when stable periodic solutions (self-oscillations/stable limit cycle) emerge from a transition through the bifurcation point. Now, the boundary of stability is called ‘safe’. It is seen that for first and second fixed points we have: r(1) = a + b, R(1) = −a; r(2) = −a − b; R(2) = a + 2b. According to [12,23,40,41], we obtain the formula describing the first Lyapunov value for our system and calculate its value in the calculated bifurcation points (for more details, see AppendixA). Thus, for L1, we write:

πa2 h i L (λ ) = bc b2 + 6 − 3b2 − 8 , (12) 1 0 2 2 2b(b + 4)∆0

where ∆0 = 1 − bc, and λ0 is defined as a value of a and b for which the relation R = 0 (1) takes place. It is clear that: (i) for a = 0, b > 0, then R = 0 and L1 = 0, i.e., the (2) first fixed point O1 is structurally unstable; (ii) for a = −2b, b > 0, then R = 0 and 2πb 4 2 L1 = − 2 b + 9b + 8 < 0. Hence, according to [9,10,12], a soft stability loss (b2+4)(1+b2) takes place, i.e., in the case of transition through the AH boundary (R = 0) from positive values to negative ones, a stable limit cycle (self-oscillations) emerges. Conversely, in the case of a transition from negative values to positive ones the stable limit cycle disappears, i.e., the self-oscillations cease. In the following Section3, we demonstrate numerically this type of behavior.

2.2. Special Cases 2.2.1. Conservative Case It is well-known that conservative systems play an important role in many mathematical and mechanical problems [11,42,43]. The equations of motion of such systems can be Mathematics 2021, 9, 352 5 of 17

obtained from their Hamiltonian. A one-dimension system, with position coordinate y, momentum x and Hamiltonian H(x, y) has equations of motion

. ∂H(x, y) . ∂H(x, y) y = , x = − . (13) ∂x ∂y

Here H(x, y) is a ﬁrst integral because

. ∂H . ∂H d x + y = H ≡ 0, (14) ∂x ∂y dt

i.e., H remains constant along the trajectories. In other words, H is a conserved quantity (constant of the motion). It is important to note that the existence of a ﬁrst integral is useful because of the connection between its level curves and the trajectories of the system. For a = b = 0, the system (1) reduces to a planar R2 Hamiltonian system with Hamiltonian (ﬁrst integral)

x2 y2 H(x, y) = + − Cy = h, (15) 2 2 where C is a real constant. This integrable system has a fixed point O(0, −C) of center type, i.e., the trajectories in a neighborhood of fixed point are periodic (if C = 0), or of saddle type (if C 6= 0). Note here that, the following center forms exist: linear type center, nilpotent center and degenerate center. If the fixed point is of a focus or a center type, we say that it is monodromic singular point [44]. When C = 1, the equilibrium state O is a 3 saddle at the level H = 2 . It is easy to see that, in this case, the general solution reeds

x(t) = c1 cos(t) − c2 sin(t), y(t) = c2 cos(t) + c1 sin(t) − c3, (16) z(t) = c3,

where c1, c2, and c3 are real constants.

2.2.2. Dissipative Case Let us view the situation when a = 0 and b > 0. Then for system (1) we have

. x = −y − z, . y = x, (17) . z = −bz,

i.e., we obtain a linear 3D autonomous system. It is seen that the last equation into (17) contains only the variable z. Thus, after a direct calculation, we derive from the third equation of system (17) that for z 6= 0,

−bt |z| = c3e , (18)

where |−bt| < +∞. On the other hand, the differential equation

dx −y − z = , x 6= 0 (19) dy x

has solutions on the plane x2 y2 + + yc e−bt = h , (20) 2 2 3 1

where h1 is a real constant. Note that the module of z is strictly bounded below by b > 0 and t → +∞ , i.e., z(t) satisﬁes |z| → 0. Mathematics 2021, 9, 352 6 of 17

A noteworthy, easily derivable analytical solution is this:

x(t) = c (t) − c (t) − c 1 (b (t) + (t) − b (−bt)) 1 cos 2 sin 3 1+b2 cos sin exp , y(t) = c (t) + c (t) − c 1 (b (t) − (t) + b (−bt)) (21) 2 cos 1 sin 3 1+b2 sin cos exp , Mathematics 2021, 9, x FOR PEER REVIEW 7 of 19 z(t) = c3 exp(−bt),

where similar to the previous case c1, c2, and c3 are real constants. 2.3. Global Analysis 2.3. Global Analysis These bifurcations for which it is insufficient to consider small local surroundings These bifurcations for which it is insufficient to consider small local surroundings near the equilibrium states or the limit cycles are called global. They lead to qualitative near the equilibrium states or the limit cycles are called global. They lead to qualitative change in the stable and the unstable manifolds of the states of equilibrium and/or the change in the stable and the unstable manifolds of the states of equilibrium and/or the limit cycles. One of these situations is the occurrence of a loop of the separatrices of one limit cycles. One of these situations is the occurrence∈[]− of a loop of the∈ separatrices( ] of andone the and same the same saddle saddle (steady (steady state) state) [14,45]. [14,45 For]. For a a ∈ 1[−.511.51,,1 1and] and bb ∈0(.290.29,, 0. 0.88 ]wewe obtainobtain [34] [34] that the twotwo fixedfixed pointspointsO O1 1and andO 2Oat2 bifurcationat bifurcation parameters parametersa and/or a and/orb are bof saddle-focusare of saddle-focustype: (i) type: unstable (i) focus-negativeunstable focus-negative real eigenvalue real eigenvalue and complex and eigenvalues complex eigenvalueswith positive with real positive part (see real Figure part 1(see left Figure panel); 1 (ii)left stable panel); focus-positive (ii) stable focus-positive real eigenvalue real eigenvalueand complex and eigenvalues complex eigenvalues with negative with real negati partve (see real Figure part1 (seeright Figure panel). 1 Noteright thatpanel). for some subintervals the fixed point O can be of stable focus type. These two fixed points Note that for some subintervals the fixed2 point O2 can be of stable focus type. These two fixedcan be points included can in be homoclinic/heteroclinic included in homoclinic/heteroclinic structures of Shilnikov structures type, of where Shilnikov their stabletype, and unstable invariant manifolds (Ws and Wu), are meeting each other in a most intricate where their stable and unstable invariant manifolds (W s and W u ), are meeting each manner. The structure of phase space near homoclinic loop essentially depends on the other in a most intricate manner. The structure of phase space near homoclinic loop saddle index δ, i.e., when δ > 1, a simple dynamics takes place, and when δ < 1, a complex essentially depends on the saddle index δ , i.e., when δ >1, a simple dynamics takes dynamics can be seen. place, and when δ <1 , a complex dynamics can be seen.

W s W u

O1 O2 W s W u

FigureFigure 1. 1. DepictedDepicted are are the the fixed ﬁxed points points of of saddle-focus saddle-focus type of system (1). Left panel: panel :unstable unstable focus focus (complex(complex eigenvalues eigenvalues with with positive positive real real part). part). RiRightght panel: panel :stable stable focus focus (complex (complex eigenvalues eigenvalues with with negative real part). negative real part).

ItIt is is well-known thatthat the the existence existence of of a homoclinic/heteroclinica homoclinic/heteroclinic cycle cycle is one is one of the of com-the commonmon manners manners of the of the formation formation or disappearance or disappearan ofce a of limit a limit cycle, cycle, when when the the limit limit cycle cycle em- emanatesanates from, from, or approachesor approaches the heteroclinic the heteroclinic cycle ascycle a singular as a singular limit, respectively limit, respectively [14,45,46] . [14,45,46].There are knownThere are situations knownin situations which a uniquein which limit a unique cycle is limit born, cycle and certainis born, criteria and certain can be criteriaused to can determine be used if to this determine limit cycle if mustthis limit be stable cycle or must unstable. be stable As we or mentionedunstable. As above, we mentionedthe system above, (1) has the steady system states (1) ofhas saddle-focus steady states type. of saddle-focus Therefore, in type. our caseTherefore, the Shilnikov in our casetheory the for Shilnikov analyzing theory of bifurcations for analyzing of homoclinic/heteroclinic of bifurcations of trajectorieshomoclinic/heteroclinic and existence trajectoriesof complex and dynamics existence (chaos) of complex in ﬂow dynamics in R3 can (chaos) be used in flow [24]. in In R this3 can connection, be used [24]. in theIn thisAppendix connection,B, we in explain the Appendix shortly the B, generalwe expl resultsain shortly obtained the general by Shilnikov. results Onobtained the other by Shilnikov.hand, historically, On the other the four-dimensional hand, historically, case the was four-dimensional considered in 1967 case [ 47was], and considered the general in 1967case-three [47], and years the later general in [25 case-three]. years later in [25]. ForFor the the system system (1), (1), if: if: (1) (1) aa==−−1.1.5,5, b b==0.60.6, (2), (2) aa==−−1,1,b b==0.350.35 andand (3) (3) aa = bb==00.5.5 thethe equilibriums equilibriums (fixed (ﬁxed points) points) an andd their their eigenvalues eigenvalues are are given given by: by:

Case 1. O1(0, 0, 0), then (χ1, χ2, χ3) = (0.5508, −0.5754 ± 1.1414i), 0.5 < δ = 0.957 < 1; () ()(χ χ χ = − ± ) Case 1. O1 0, 0, 0 , then 1 , 2 , 3 0.5508, 0.5754 1.1414i , O2(0, 0.6, −0.6), then (χ1, χ2, χ3) = (−0.7855, 0.0928 ± 1.0664i), δ = 0.1181 < 0.5. 0.5 < δ = 0.957 <1;

()− ()(χ χ χ = − ± )δ = < O2 0, 0.6, 0.6 , then 1 , 2 , 3 0.7855, 0.0928 1.0664i , 0.1181 0.5 .

() ()(χ χ χ = − ± )δ = > Case 2. O1 0, 0, 0 , then 1 , 2 , 3 0.4694, 0.4097 1.1031i , 1.146 1;

Mathematics 2021, 9, 352 7 of 17

Case 2. O1(0, 0, 0), then (χ1, χ2, χ3) = (0.4694, −0.4097 ± 1.1031i), δ = 1.146 > 1;

O2(0, 0.65, −0.65), then (χ1, χ2, χ3) = (−0.5754, 0.1127 ± 1.0569i), δ = 0.196 < 0.5.

Case 3. O1(0, 0, 0), then (χ1, χ2, χ3) = (−0.8038, 0.1519 ± 1.105i), δ = 0.189 < 0.5;

O2(0, 2, −2), then (χ1, χ2, χ3) = (0.6015, −0.5507 ± 1.1659i), δ = 1.092 > 1.

Since the complex exponents χ2,3 for the saddle-foci O2 (in Cases 1 and 2) and O1 (in Case 3) are nearest to the imaginary axis, the homoclinic loop implies the emergence of infinitely many periodic orbits of saddle type. Moreover, since the second saddle value σ1 (see AppendixB) is negative, then near the homoclinic loop may exist stable periodic orbits along with saddle ones. Note that these stable orbits are with long periods and have weak attraction basins, i.e., they are practically invisible in numerical experiments. Interestingly, when the equilibrium states O1 (in Cases 1 and 2) and O2 (in Case 3) u s have exponents (dimW = 1, dimW = 2), the second saddle value σ1 is also negative and therefore in a small neighborhood of the homoclinic loop there are stable periodic orbits. Moreover, the condition 0.5 < δ < 1 in this case is necessary to exclude the transition to complex dynamics via codimension-two bifurcations of homoclinic loop(s) [9,14,48]. Note here that these bifurcations of homoclinic loops are essential for bifurcation behavior in the Lorenz system [30]. Finally, we will remark the following fact: at the non-trivial equilibrium O2, the homoclinic bifurcations always started/finished from the super-critical Andronov-Hopf bifurcation (the first Lyapunov value is negative).

3. Numerical Investigation In the considered case here, some of the known analytical techniques and results are not applicable [16,49,50], and we are forced to use numerical calculations and specific features of system (1). Hence, in this section we find some new results for its bifurcation behavior and route to chaos. In order to compare the analytical predictions with numerical results, the governing equations of system (1) were solved numerically using MATLAB (The MathWorks, Inc., Natick, MA, USA). Consider the numerical technique (based on the initial value problem) used here, we note that it does not allow to detect and numerically trace periodic orbits and predict their bifurcations. In this case, a possibility is to use the systematic numerical approach presented in [51]. Figure2 shows bifurcation diagrams for the system (1): (a) and (c) values of y coordinate versus b; and (b) and (d) values of z coordinate are plotted against b regarded as a continuously varying control (bifurcation) parameter, when the parameter a = −1.51 is fixed. It is seen that for b ∈ [0.495, 0.524] the system is chaotic. On a further increase of the bifurcation parameter b, the system (1) exhibits inverse period-doubling bifurcations leading to a periodic motion. A more detailed investigation of the system (1) in the region b ∈ [0.52, 0.6] is shown in Figure2c,d, i.e., when the bifurcation parameter b varies in narrower ranges. Practically, we observe in this interval a cascade of inverse period-doubling bifurcations when the fixed points O1 and O2 are of saddle-focus type, i.e., O1 possesses a 2-dimensional stable manifold and a 1-dimensional unstable manifold, and O2 possesses a 2-dimensional unstable manifold and a 1-dimensional stable manifold. It is interesting to note that O2 becomes stable focus (dimWs = 3, dimWu = 0) after b = 0.75 and remains stable till the end of the interval b ∈ (0.75, 0.8]. In other words, as a result of the evidences shown in Figure2a,b, we can conclude that a stable limit cycle (small amplitude attenuation oscillations) with period one occurs when O2 becomes stable focus-see for details Figure3. Denote here that for b ∈ [0.77, 0.8] the system (1) has only stable solutions-the self-oscillations cease. These numerical results (calculations) are in exact accordance with the corresponding analytical results obtained for first Lyapunov value in previous Section 2.1. Mathematics 2021, 9, x FOR PEER REVIEW 9 of 19

a result of the evidences shown in Figure 2a,b, we can conclude that a stable limit cycle

(small amplitude attenuation oscillations) with period one occurs when O2 becomes stable focus-see for details Figure 3. Denote here that for b∈[]0.77, 0.8 the system (1) Mathematics 2021, 9, 352 has only stable solutions-the self-oscillations cease. These numerical results (calculations)8 of 17 are in exact accordance with the corresponding analytical results obtained for first Lyapunov value in previous Section 2.1.

(a) (b)

(c) (d) () () FigureFigure 2. 2.Bifurcation Bifurcation diagrams: diagrams: (a (,ac),cy) (ty) tversus versusb; andb ; (andb,d )(zb(,dt)) versusz t versusb, generated b , generated by computer by computer solution solution of the system of the (1)system at a = (1)− 1.51at .a The= −1 initial.51 . The conditions initial areconditionsx(0) = yare(0) =x()z0(0=) y= ()00.1= .z () Note0 = 0. that1. Noteb ∈ [ 0.495,that 0.77b ∈[]]0in.495 (a,b, );0. and77 bin∈ ([a0.52,,b); 0.6and] inb (∈c,[]d0)..52, 0.6 in (c,d).

In Figure4 the bifurcation diagrams of the system (1) when a = −1 are shown. It can be seen that at b ∈ [0.315, 0.32] chaotic solution occurs. In analogy with the previous case, the system passes from chaos to regular motion after inverse period-doubling bifurcations. In this case the ﬁxed points O1 and O2 are also of saddle-focus type, i.e., O1 possesses a 2-dimensional stable manifold and a 1-dimensional unstable manifold, and O2 possesses a 2-dimensional unstable manifold and a 1-dimensional stable manifold. Here O2 becomes stable focus after b = 0.5 and remains stable till the end of the interval b ∈ (0.75, 0.65]. It is interesting to note that here the system (1) has regular solutions at the beginning and in the end of the interval for the control parameter b. Comparing Figures2 and4 we conclude that in the case, when a = −1.51 the chaotic zone is longer than those obtained in Figure4 for a = −1. It is interesting to note that inverse period-doubling bifurcations can be seen in many experimental systems, for example triple physical pendulum [52]. Thus, it is possible to detect the hysteresis phenomena, i.e., it is expected that bifurcation diagrams presented (a) in Figure4 will be different with increase and decrease(b of) the parameter b. Figure5 shows bifurcation diagrams for the system (1): (a) and (c) values of y coordinate versus a; and (b) and (d) values of z coordinate are plotted against a regarded as a continuously varying control (bifurcation) parameter, when the parameter b = 0.5 is ﬁxed. Similar to the previous case for a = −1.51 (see Figure2), initially the system (1) is

chaotic. On a further increase of the bifurcation parameter a, the system (1) exhibits inverse period-doubling bifurcations leading to a periodic motion-stable limit cycle according to analytical results obtained for ﬁrst Lyapunov value in Section 2.1. The white zones, seen in Figure5a,b, correspond to very fast inverse period-doubling bifurcations-see Figure5c,d. Mathematics 2021, 9, x FOR PEER REVIEW 9 of 19

a result of the evidences shown in Figure 2a,b, we can conclude that a stable limit cycle

(small amplitude attenuation oscillations) with period one occurs when O2 becomes stable focus-see for details Figure 3. Denote here that for b∈[]0.77, 0.8 the system (1) has only stable solutions-the self-oscillations cease. These numerical results (calculations) are in exact accordance with the corresponding analytical results obtained for first Lyapunov value in previous Section 2.1.

(a) (b)

Mathematics 2021, 9, x FOR PEER REVIEW 10 of 19

(c) (d) Figure 2. Bifurcation diagrams: (a,c) y()t versus b ; and (b,d) z()t versus b , generated by computer solution of the system (1) at a = −1.51 . The initial conditions are x()0 = y ()0 = z ()0 = 0.1. Note that b ∈[]0.495, 0.77 in (a,b); and Mathematics 2021, 9, 352 9 of 17 b ∈[]0.52, 0.6 in (c,d).

(c) (d) Mathematics 2021, 9, x FOR PEER REVIEW 10 of 19 Figure 3. Stable limit cycle for system (1) (for coordinate x (a); for coordinate y (b), for coordinate z (c) and phase space ()2 (d)) when a = −1.51 and b = 0.75 . Note that for these values of a and b , R = −0.01. (a) (b) In Figure 4 the bifurcation diagrams of the system (1) when a = −1 are shown. It can be seen that at b ∈[]0.315,0.32 chaotic solution occurs. In analogy with the previous case, the system passes from chaos to regular motion after inverse

period-doubling bifurcations. In this case the fixed points O1 and O2 are also of

saddle-focus type, i.e., O1 possesses a 2-dimensional stable manifold and a

1-dimensional unstable manifold, and O2 possesses a 2-dimensional unstable manifold = and a 1-dimensional stable manifold. Here O2 becomes stable focus after b 0.5 and remains stable till the end of the interval b∈(0.75, 0.65]. It is interesting to note that here the system (1) has regular solutions at the beginning and in the end of the interval for the control parameter b . Comparing Figure 2,4 we conclude that in the case, when a = −1.51 the chaotic zone is longer than those obtained in Figure 4 for a = −1. It is (c) interesting to note that inverse period-doubling bifurcations(d) can be seen in many experimental systems, for example triple physical pendulum [52]. Thus, it is possible to FigureFigure 3. Stable 3. Stable limit cycle limit forcycle system for system (1) (for (1) coordinate (for coordinatex (a); for x coordinate(a); for coordinatey (b), for y coordinate (b), for coordinatez (c) and phasez (c) and space phase (d)) space detect the hysteresis phenomena, i.e., it is expected()2 that bifurcation diagrams presented when (ad=)) when−1.51 anda = −b1=.510.75. and Note b = 0 that.75 for. Note these that values for these of a valuesand b, Rof(2 )a= and−0.01. b , R = −0.01. in Figure 4 will be different with increase and decrease of the parameter b . In Figure 4 the bifurcation diagrams of the system (1) when a = −1 are shown. It can be seen that at b ∈[]0.315,0.32 chaotic solution occurs. In analogy with the previous case, the system passes from chaos to regular motion after inverse

period-doubling bifurcations. In this case the fixed points O1 and O2 are also of

saddle-focus type, i.e., O1 possesses a 2-dimensional stable manifold and a

interesting to note that inverse period-doubling bifurcations can be seen in many (a) (b) experimental systems, for example triple physical pendulum [52]. Thus, it is possible to () () FigureFigure 4. Bifurcation 4. Bifurcation diagrams: diagrams: (adetect) y( t(a))versus they t hysteresis versusb; and (bb); zandphenomena,(t) versus(b) z tb, generatedversus i.e., it bis, bygeneratedexpected computer bythat solution computer bifurcat of solution theion system diagrams of the (1) atsystem presented = − ()= ()= ()= ∈[] a = −(1)1. Theat a initial1 . conditionsThe initial areconditionsinx( 0Figure) = yare( 04 ) willx=0z( 0be)y =0different0.1.z 0 Note0 with.1 that. Note bin∈crease that[0.29, b 0.65and0]. .29decrease, 0.65 . of the parameter b .

In FigureFigure6, the5 shows Andronov-Hopf bifurcation bifurcation diagrams curvefor the (boundary system (1): of stability(a) and loss(c) valuesR = 0 )of y of thecoordinate fixed point versusO2 is showna ; and for(b) differentand (d) valuesvalues ofof thez bifurcationcoordinate are parameters plotted aagainstand a b. Followingregarded [ 9as,10 a, 12continuously,23] and results varying obtained control in (bifurcation) previous Section parameter, 2.1 for when first Lyapunov the parameter value, we conclude that this boundary of stability is “safe”, and a soft stability loss occurs. Therefore, the transformation into/from chaos is related to a soft loss of stability, i.e., the fixed point O2 is a stable weak focus on the stability boundary.

(a) (b) Figure 4. Bifurcation diagrams: (a) y()t versus b ; and (b) z()t versus b , generated by computer solution of the system (1) at a = −1 . The initial conditions are x()0 = y ()0 = z ()0 = 0.1. Note that b∈[]0.29, 0.65 .

Figure 5 shows bifurcation diagrams for the system (1): (a) and (c) values of y coordinate versus a ; and (b) and (d) values of z coordinate are plotted against a regarded as a continuously varying control (bifurcation) parameter, when the parameter

Mathematics 2021, 9, x FOR PEER REVIEW 11 of 19

b = 0.5 is fixed. Similar to the previous case for a = −1.51 (see Figure 2), initially the system (1) is chaotic. On a further increase of the bifurcation parameter a , the system (1) exhibits inverse period-doubling bifurcations leading to a periodic motion-stable limit Mathematics 2021, 9, 352 cycle according to analytical results obtained for first Lyapunov value in Section10 of 2.1. 17 The white zones, seen in Figure 5a,b, correspond to very fast inverse period-doubling bifurcations-see Figure 5c,d.

(a) (b)

(c) (d) Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19 FigureFigure 5. Bifurcation 5. Bifurcation diagrams: diagrams: (a,c) y( t()a,versusc) y()t a ;versus and (b ,da);z and(t) versus (b,d) az,() generatedt versus bya , computergenerated solution by computer of the systemsolution (1) of the at b =system0.5. The (1) initial at b conditions= 0.5 . The areinitialx(0 conditions) = y(0) = arez(0 )x=()0 0.1.= y () Note0 = z ()that0 = 0a.1∈. [−Note1.51, that−1.03 a ]∈. []−1.51, −1.03 .

In Figure 6, the Andronov-Hopf bifurcation curve (boundary of stability loss R = 0 )

of the fixed point O2 is shown for different values of the bifurcation parameters a and b . Following [9,10,12,23]R=0 and results obtained in previous Section 2.1 for first Lyapunov value, we conclude that this boundary of stability is “safe”, and a soft stability loss occurs. Therefore, the transformationR>0 into/from chaos is related to a soft loss of stability, i.e., the fixed point O is a stable weak focus on the stability boundary. 2 (stable zone)

R<0 (unstable zone)

∈[] FigureFigure 6. 6. BifurcationBifurcation diagram diagram of ofsystem system (1) (1) at atfixed ﬁxed point point OO2 ,2 when, when and and bb ∈ 0[0.05,.05, 0 0.755.755].. Here Here we wenote note that thatR =R0= is0 boundary is boundary for Andronov-Hopffor Andronov-Hopf bifurcation bifurcation (AH). (AH).

In Figure7 we show the bifurcation diagrams of the system (1) when a ∈ [0.1, 0.51] In Figure 7 we show the bifurcation diagrams of the system (1) when a∈[]0.1, 0.51 and b = 0.5. In this case the system (1) demonstrates a period-doubling route to chaos-a and b = 0.5. In this case the system (1) demonstrates a period-doubling route to chaos-a steady state loses its stability and simultaneously a new orbit of doubled period is created. steady state loses its stability and simultaneously a new orbit of doubled period is Here the ﬁxed points O1 and O2 are of saddle-focus type in all interval for bifurcation created. Here the fixed points O and O are of saddle-focus type in all interval for parameter a, i.e., O1 possesses a1 2-dimensional2 unstable manifold and a 1-dimensional bifurcation parameter a , i.e., O1 possesses a 2-dimensional unstable manifold and a

1-dimensional stable manifold, and O2 possesses a 2-dimensional stable manifold and a 1-dimensional unstable manifold. It is interesting to note that when a = 0.51 and b = 0.5, the system has pseudo-chaotic (order in chaos) behavior for very short intervals of time-see for details Figure A4.

(a) (b)

Mathematics 2021, 9, x FOR PEER REVIEW 12 of 19

R=0

R>0 (stable zone)

R<0 (unstable zone)

∈[] Figure 6. Bifurcation diagram of system (1) at fixed point O2 , when and b 0.05, 0.755 . Here we note that R = 0 is boundary for Andronov-Hopf bifurcation (AH).

In Figure 7 we show the bifurcation diagrams of the system (1) when a∈[]0.1, 0.51 and b = 0.5. In this case the system (1) demonstrates a period-doubling route to chaos-a Mathematics 2021, 9, 352 steady state loses its stability and simultaneously a new orbit of doubled period11 of 17is

created. Here the fixed points O1 and O2 are of saddle-focus type in all interval for

bifurcation parameter a , i.e., O1 possesses a 2-dimensional unstable manifold and a

stable1-dimensional manifold, stable and Omanifold,2 possesses and a 2-dimensionalO2 possesses a stable 2-dimensional manifold stable and a manifold 1-dimensional and a unstable1-dimensional manifold. unstable It is interesting manifold. to It note is interesting that when ato= 0.51noteand thatb =when0.5, thea = system0.51 and has pseudo-chaoticb = 0.5, the system (order has in pseudo-chaotic chaos) behavior (order for very in chaos) short behavior intervals for of time-seevery short for intervals details Figureof time-see A4. for details Figure A4.

Mathematics 2021, 9, x FOR PEER REVIEW 13 of 19

(a) (b)

Figure 7.7. BifurcationBifurcation diagrams:diagrams: ( a(,ac,)cy) (ty)()tversus versusb; andb ; (andb,d )(zb(,td)) versusz()t versusa, generated a , generated by computer by computer solution ofsolution the system of the (1) systemat b = 0.5. (1) at The b initial= 0.5 . conditions The initial are conditionsx(0) = y are(0) =x()0z(0=)y= ()0 0.1.= z () Note0 = 0. that1. Notea ∈ [that0.1, 0.51a ∈].[]0.1, 0.51 .

Finally,Finally, discussing the results shown in Figu Figurere 55c,d,c,d, FigureFigure7 c,d,7c,d, it it is is seen seen that that an an apparent sudden collapse/appearance in the chaotic attractor size occurs at a value of apparent sudden collapse/appearance in the chaotic attractor size occurs at a value of control parameter a ≈ −1.464 (or a ≈ 0.465). According to [53–56], such a sudden control parameter a ≈ −1.464 (or a ≈ 0.465 ). According to [53–56], such a sudden qualitative change in a chaotic attractor is called interior crisis. It is in a good agreement qualitative change in a chaotic attractor is called interior crisis. It is in a good agreement with our analytical results obtained in Section2, the Theorem explained in AppendixB with our analytical results obtained in Section 2, the Theorem explained in Appendix B and the numerical results in the AppendixC—see Figures A1 and A2. and the numerical results in the Appendix C—see Figures A1 and A2. 4. Conclusions 4. ConclusionThe papers presents a study of the dynamic behavior of the so-called Rössler prototype-4 systemThe, using paper analytical presents and a numerical study of tools. the Thedynamic governing behavior differential of the equations so-called of systemRössler prototype-4(1) were solved system numerically, using analytical using MATLAB and numerical (The MathWorks, tools. The Inc., governing Natick, MA, differential USA). For equationsall simulations of system the initial (1) were conditions solved numeri were xcally(0) = usingy(0) =MATLABz(0) = 0.1(The. The MathWorks, analytical Inc., and Natick,numerical MA, results leadUSA). to theFor following all comments:simulations (1) thethe systeminitial (1) hasconditions two fixed pointswere ofx()0 saddle-focus= y ()0 = z ()0 type,= 0.1 and. The therefore analytical homoclinic/heteroclinic and numerical results structures lead to of Shilnikovthe following type comments:take place; (2)(1) forthe values system of the(1) coefficientshas two fixeda and pobintsdifferent of saddle-focus from these type, in [4], and the systemtherefore (1) homoclinic/heteroclinichas chaotic solutions; (3) structures the original of systemShilnikov (1) cantype be take presented place; in(2) the for form values of aof linear the coefficientsoscillator with a oneand nonlinear b different automatic from these regulator; in [4], (4) the the system route (1) chaos has starts/finisheschaotic solutions; from (3) a a = b = thesoft original (reversible) system loss of(1) stability; can be (5)presented for 0.51 in andthe form0.5 of, thea linear system oscillator has pseudo-chaotic with one (order in chaos) behavior for very short intervals. nonlinear automatic regulator; (4) the route chaos starts/finishes from a soft (reversible) loss of stability; (5) for a = 0.51 and b = 0.5, the system has pseudo-chaotic (order in chaos) behavior for very short intervals. Generalizing the numerical results shown in Figures 2–7 and those in Appendix C for Rössler prototype-4 system (1), we can conclude that: (i) for values of bifurcation parameter b∈[]0.495,0.524 (for a = −1.51), the system is in a chaotic regime. In result of inverse period-doubling bifurcations, for values of b > 0.54 , the system has only regular solutions with different period; (ii) for values b∈[]0.315,0.32 (when a = −1), the system (1) is in a chaotic state, as for b∈(0.32,0.65] period-doubling bifurcations take place. For b = 0.5, a stable limit cycle emerges whose period is one; iii) for values of bifurcation parameter a∈[]−1.51,−1.4 or a ∈[]0.41,0.51 (for b = 0.5), the system (1) has chaotic/regular solutions. In these intervals for a , very fast inverse period-doubling bifurcations and so-called interior crisis take place.

Mathematics 2021, 9, 352 12 of 17

Generalizing the numerical results shown in Figures2–7 and those in AppendixC for Rössler prototype-4 system (1), we can conclude that: (i) for values of bifurcation parameter b ∈ [0.495, 0.524] (for a = −1.51), the system is in a chaotic regime. In result of inverse period-doubling bifurcations, for values of b > 0.54, the system has only regular solutions with different period; (ii) for values b ∈ [0.315, 0.32] (when a = −1), the system (1) is in a chaotic state, as for b ∈ (0.32, 0.65] period-doubling bifurcations take place. For b = 0.5, a stable limit cycle emerges whose period is one; iii) for values of bifurcation parameter a ∈ [−1.51, −1.4] or a ∈ [0.41, 0.51] (for b = 0.5), the system (1) has chaotic/regular solutions. In these intervals for a, very fast inverse period-doubling bifurcations and so-called interior crisis take place.

Author Contributions: The initial idea for investigation comes from S.G.N. Stability and bifurcation analysis were performed by S.G.N. The analytical solutions were obtained from V.M.V. and S.G.N. The manuscript was drafted by all coauthors. All authors have read and agreed to the published version of the manuscript. Funding: The author V.M. Vassilev gratefully acknowledges the financial support via contract H 22/2 with Bulgarian National Science Fund. Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of NAME of Institute of Mechanics-BAS. Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Data Availability Statement: Data supporting reported results can be found in SCOPUS (https:// www.scopus.com/search/form.uri?display=basic#basic (accessed on 5 January 2021)) and Web of science (https://apps.webofknowledge.com/WOS_GeneralSearch_input.do?product=WOS&search_ mode=GeneralSearch&SID=D5mCh5CCu8VaTeN1Hdr&preferencesSaved= (accessed on 5 January 2021)). Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

Analytical calculation of ﬁrst Lyapunov value—L1(λ0) In the case of three nonlinear ODEs, the ﬁrst Lyapunov value can be determined analytically by the formula in [12]:

π h (2) (3) (2) (3) (2) (2) (2) L1(λ0) = 4q 2 A33 A33 − A22 A22 + 2A23 A22 + A33 − (3) (3) (3) √ (2) (3) (2) (3) i −2A23 A22 + A33 + 3 q A222 + A333 + A233 + A223 + n h (1) (2) (3) (1) (2) (3) + √ π p2 A A + A + A A + A + 4p q(p2+4q) 2 22 3 12 13 2 33 12 3 13 (A1) (1) (2) (3)i √ h (1) (1) (2) (3) + 4A23 A13 + A12 + 4p q A22 − A33 A13 + A12 + (1) (3) (2)i (1) (1) (2) (3)o + 2A23 A13 − A12 + 16q A22 + A33 A12 + A13 ,

where ∆0 = 1 − bc, and λ0 is defined as a value of a and b for which the relation R = 0 n n takes place. The coefficients Aij and Aijk (i, j, k, n = 1, 2, 3) are defined by corresponding formulas presented in [12]. After accomplishing some transformations and algebraic operations for the coefficients n n Aij and Aijk, we obtain:

( ) ( ) ( ) ( ) A 1 = 1 α0 a, A 2 = A 2 = 1 α0 a, A 3 = A0 = 1 α0 a, 22 ∆0 11 22 12 ∆0 21 22 31 ∆0 31 (1) (2) (3) (1) (2) (2) (3) (2) (3) (2) (3) A33 = A33 = A33 = A23 = A23 = A13 = A13 = A222 = A333 = A233 = A223 = 0. (A2) Mathematics 2021, 9, 352 13 of 17

Here, we note that

α11 α12 α13

∆ = det α α α = 1 − bc, (A3) 0 21 22 23 α31 α32 α33

where α = −bc, α = c, α = α = 1, 11 21 31 32 (A4) α12 = α23 = −1, α22 = α13 = α33 = 0. From the previous formula (A3), we obtain that

0 0 0 α11 = α22α33 − α23α32 = 1, α21 = α21α33 − α23α31 = −1, α31 = α21α32 − α22α31 = c (A5)

Hence, after substitution of (A2) and (A5) into (A1), for ﬁrst Lyapunov value we have:

πa2 h i L (λ ) = bc b2 + 6 − 3b2 − 8 . (A6) 1 0 2 2 2b(b + 4)∆0

Appendix B Shilnikov criteria for chaos We consider here a three dimensional autonomous system

. dx x = = f (x, µ), x ∈ R3, µ ∈ R1 (A7) dt where the nonlinearity f is sufﬁciently smooth for the results to hold. It is well-known that if the system (A7) has ﬁxed points of saddle-focus type, then the Shilnikov theory for analyzing of bifurcations of homoclinic/heterclinic orbits and existence of chaos can be used [11,24,25,57,58]. The Shilnikov criteria for saddle-focus equilibria can be summarized in the following theorem [20,58,59]:

Theorem A1. Suppose that system (A7) has at µ = 0 a saddle-focus equilibrium point x0 = 0 with eigenvalues χ1(0) > 0 > Reχ2,3(0) and a homoclinic orbit Γ0. Assume the following genericity conditions:

Hypothesis 1 (H1). σ0 = χ1(0) + Reχ2,3(0) < 0;

Hypothesis 2 (H2). χ2(0) 6= χ3(0);

Hypothesis 3 (H3). β0(0) 6= 0, whereβ(µ) is the split function;

Hypothesis 4 (H4). σ0 = χ1(0) + Reχ2,3(0) > 0.

Hence, (i) if the conditions (H1–H3) are valid then system (A7) has a unique and stable limit cycle Lβ in a neighborhood U0 of Γ0 ∪ x0 for all sufficiently small β > 0; (ii) if the conditions H2 and H4 are valid then system (A7) has an infinite number of saddle limit cycles in a neighborhood U0 of Γ0 ∪ x0 for all sufficiently small |β|.

Remark (about part (i) of Theorem A1). 1. For all sufficiently small β ≤ 0 (where the split function is a functional defined on the original and perturbed systems), the system (A7) has no u periodic orbits in U0 and the unstable manifold W (x0) tends to the cycle Lβ; 2. For β = 0, the cycle period tends to infinity.

Remarks (about part (ii) of Theorem A1). 1. For β taking positive or negative values, an inﬁnite number of bifurcations occur. Some of these bifurcations are related to a “basic” limit cycle; Mathematics 2021, 9, x FOR PEER REVIEW 16 of 19 Mathematics 2021, 9, x FOR PEER REVIEW 16 of 19

Mathematics 2021, 9, 352 2. The “basic” cycle disappears and appears via tangent/fold bifurcation infinity 14many of 17 times. Moreover,2. The “basic” the cycle cycle also disappears exhibits anand infinit appearse number via tangent/fold of period-doubling bifurcatio bifurcations;n infinity many times. 3.Moreover, The “basic” the cycle cycle also and exhibits the secondary an infinit cyclese number gen oferated period-doubling by period-doublings bifurcations; can be stable or 3. The “basic” cycle and the secondary cycles generated by period-doublings can be stable or 2. Therepelling, “basic” cycledepending disappears on the andsign appearsof the diverg via tangent/foldence of (A7) at bifurcation the saddle-focus, inﬁnity i.e., many times. repelling, depending on the sign of the divergence of (A7) at the saddle-focus, i.e., Moreover, the cycle also exhibits an inﬁnite number of period-doubling bifurcations; σ = ()()= χ + χ 3. The “basic” cycle and the secondary1 cyclesdivf generatedx0 , 0 by1 period-doublings2Re 2,3. can be stable or (A8) σ = ()()divf x , 0 = χ + 2Re χ . repelling, depending on the sign of the divergence1 of (A7)0 at the1 saddle-focus,2,3 i.e., (A8) σ < β If 1 0 the “basic” cycle near the bifurcation is stable only in short intervals of . If σ < 0 σ1 = (div f )(x0, 0) = χ1 + 2Reχ2,3. (A8) β σ >If 1 the “basic” cycle near the bifurcation is stable only in short intervals of . If 1 0 there are intervals where the “basic” cycle is absolutely unstable (repelling); σ > 0 there are intervals where the “basic” cycle is absolutely unstable (repelling); If4.σ 11For< 0βthe→ “basic”0 ()β > cycle0 , nearthe system the bifurcation (A7) has is double stable onlyhomoclin in shortic loops intervals with ofdifferentβ. If σ 1(increasing)> 0 βi → ()βi > there are4. For intervals i where0 thei “basic”0 , the system cycle is (A7) absolutely has double unstable homoclin (repelling);ic loops with different (increasing) number of rotations near the saddle-focus. 4. Fornumberβi → 0( ofβi rotations> 0), the near system the saddle-focus. (A7) has double homoclinic loops with different (increasing) number of rotations near the saddle-focus. s u Initially it was assumed that n− = dimW = 2 and n+ = dimW =1. To apply the = s = = u = Initially it was assumed that n− dims W 2 and n+ dimu W 1. To apply the Initially it was assumed that n− = dimn W=1=, 2nand= 2n+ = dimW = 1. To apply the above results in the opposite case— − = + = , we have to reverse the direction of above results in the opposite case—=n− 1, =n+ 2 , we have to reverse the direction of abovetime. results Thus, in the the opposite following case— substitutionsn− 1, nare+ valid:2, we χ have→ − toχ reverse, σ → the−σ direction and “stable” χ j → −χ j σi → −σi of time.time. Thus, Thus, the the following following substitutions substitutions are are valid: valid:χj →j −χj, σij ,→ −i σi andi “stable”and “stable” → → “repelling”. “repelling”.→ “repelling”. Appendix C AppendixAppendix C C

(a) (b) (a) (b) σ O O Figure A1. Results for σ1 of fixed point 1 (a) and fixed point 2 (b), as functions of control (bifurcation) parameters FigureFigure A1. Results A1. Results for σ forof fixed1 of point fixedO point(a) andO1 fixed(a) and point fixedO point(b), asO2 functions (b), as functions of control of control (bifurcation) (bifurcation) parameters parameters a ∈[]−1.51,1 and1 b ∈[]0.29, 0.8 . 1According to TheoremA12 (see Appendix B), the system has an infinite number of a ∈ [−a1.51,∈[]− 11] .and51,1b and∈ [0.29, b ∈ 0.8[]0.]29. According, 0.8 . According to TheoremA1 to TheoremA1 (see Appendix (see AppendixB), the systemB), the system has an infinitehas an infinite number number of of period-doubling bifurcations. Note that some of these bifurcations are related to a “basic” limit cycle. period-doublingperiod-doubling bifurcations. bifurcations. Note that Note some that of some these of bifurcations these bifurcations are related are related to a “basic” to a “basic” limit cycle. limit cycle.

(a) (b) (a) (b) Figure A2. Results for δ (the saddle-focus index) of ﬁxed point O1 (a) and ﬁxed point O2 (b), as functions of control (bifurcation) parameters a ∈ [−1.51, 1] and b ∈ [0.29, 0.8].

MathematicsMathematics 2021 2021, 9, ,x 9 FOR, x FOR PEER PEER REVIEW REVIEW 17 of17 19 of 19

Figure A2. Results for δ δ (the saddle-focus index) of fixed point O (a) and fixed point O (b), as functions of control Figure A2. Results for (the saddle-focus index) of fixed point 1 O1 (a) and fixed point 2 O2 (b), as functions of control Mathematics 2021, 9, 352 []−∈ ∈[] 15 of 17 (bifurcation)(bifurcation) parameters parameters a a []−∈ 1,51.1 1,51.1 and and b b ∈[]8.0,29.0 .8.0,29.0 .

(a)( a) (b)( b) == FigureFigureFigure A3. A3.Chaotic A3. Chaotic Chaotic solution solution solution (a )(a and) (anda) strange and strange strange attractor attractor attractor (b )(b for) ( forb system) for system system (1), (1), when (1), when whena = b =ba 0.5.ba == 5.0 . 5.0 .

(a)( a) (b)( b) == FigureFigureFigure A4. A4.Pseudo-chaotic A4. Pseudo-chaotic Pseudo-chaotic (order (order (order in chaos)in chaos)in chaos) solution solution solution (a) ( anda) (anda) strange and strange strange attractor attractor attractor (b )( forb) ( forb system) for system system (1), (1), when (1), when whena = 0.51, b =ba 0.5.ba == 5.0,51.0 . 5.0,51.0 . References ReferencesReferences Springer Ser. Synerg. 3 1. 1. Rössler,Rössler, O. ChaosO. Chaos and and strange strange attractors attractors in chemicalin chemical kinetics. kinetics. Springer Ser. Synerg.1979 1979, ,, 107–113. 3, 107–113. 2. Arneodo,1. Rössler, A.; Coullet, O. Chaos P.; and Tresser, strange C. Possible attractors new in strangechemical attractors kinetics. with Springer spiral Ser. structure. Synerg. 1979Commun., 3, 107–113. Math. Phys. 1981, 79, 573–579. 2. 2. Arneodo,Arneodo, A.; A.; Coullet, Coullet, P.; P.;Tresser, Tresser, C. C.Possible Possible new new strange strange attractors attractors with with spiral spiral structure. structure. Commun. Commun. Math. Math. Phys. Phys. 1981 1981, 79, ,79, [CrossRef573–579.] 3. Gurel,573–579. D.; Gurel, O. Oscillations in Chemical Reactions; Springer: New York, NY, USA, 1983. 3. Gurel, D.; Gurel, O. Oscillations in Chemical Reactions; Springer: New York, NY, USA, 1983. 4. Sprott,3. Gurel, J. Elegant D.; Gurel, Chaos. O. Algebraically Oscillations Simple in Chemical Chaotic Reactions Flows; World; Springer: Scientific: New Singapore,York, NY, USA, 2010. 1983. 4. Sprott, J. Elegant Chaos. Algebraically Simple Chaotic Flows; World Scientific: Singapore, 2010. 5. Islam,4. Sprott, M.; Islam, J. Elegant N.; Nikolov, Chaos. Algebraically S. Adaptive Simple control Chaotic and synchronizationFlows; World Scientific: of Sprott Singapore, J system 2010. with estimation of fully unknown 5. Islam, M.; Islam, N.; Nikolov, S. Adaptive control and synchronization of Sprott J system with estimation of fully unknown 5. Islam, J.M.; Theor. Islam, Appl. N.; Mech.Nikolov,2015 S.45 Adaptive control and synchronization of Sprott J system with estimation of fully unknown parameters.parameters. J. Theor. Appl. Mech. 2015, , ,45 43–56., 43–56. [CrossRef] 6. Awrejcewicz,parameters. J. Bifurcation J. Theor. Appl. and Chaos Mech. in 2015 Simple, 45 Dynamical, 43–56. Systems; World Scientific: London, UK, 1989. 6. Awrejcewicz, J. Bifurcation and Chaos in Simple Dynamical Systems; World Scientific: London, UK, 1989. 7. Awrejcewicz,6. Awrejcewicz, J. Bifurcation J. Bifurcation and Chaos: and Chaos Theory in and Simple Applications Dynamical; Springer: Systems; Berlin,World Germany,Scientific: 1995.London, UK, 1989. 7. Awrejcewicz, J. Bifurcation and Chaos: Theory and Applications; Springer: Berlin, Germany, 1995. 8. Nikolov,7. Awrejcewicz, S.; Nedkova, J. Bifurcation N. Gyrostat and model Chaos: regular Theory and and chaoticApplications behaviour.; Springer:J. Theor. Berlin, Appl. Germany, Mech. 2015 1995., 45, 15–30. [CrossRef] 8. Nikolov, S.; Nedkova, N. Gyrostat model regular and chaotic behaviour. J. Theor. Appl. Mech. 2015, 45, 15–30. 9. Shilnikov,8. Nikolov, L.; Shilnikov, S.; Nedkova, A.; Turaev,N. Gyrostat D.; Chua,model L. regularMethods and of chaotic Qualitative behaviour. Theory inJ. Theor. Nonlinear Appl. Dynamics Mech. 2015; Part, 45, II; 15–30. World Scientific: 9. Shilnikov, L.; Shilnikov, A.; Turaev, D.; Chua, L. Methods of Qualitative Theory in Nonlinear Dynamics; Part II; World Scientific: London,9. Shilnikov, UK, 2001. L.; Shilnikov, A.; Turaev, D.; Chua, L. Methods of Qualitative Theory in Nonlinear Dynamics; Part II; World Scientific: London, UK, 2001. 10. Andronov,London, A.; Witt,UK, 2001. A.; Chaikin, S. Theory of Oscillations; Addison-Wesley Reading: Boston, MA, USA, 1966. 10. Andronov, A.; Witt, A.; Chaikin, S. Theory of Oscillations; Addison-Wesley Reading: Boston, MA, USA, 1966. 11. Guckenheimer,10. Andronov, J.; A.; Holmes, Witt, A.; P. NonlinearChaikin, S. Oscillations, Theory of Oscillations Dynamical; Systems,Addison-Wesley and Bifurcations Reading: of VectorBoston, Fields MA,; Springer: USA, 1966. New York, NY, 11. Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer: New York, USA,11. Guckenheimer, 1998. J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer: New York, NY, USA, 1998. 12. Bautin,NY, N. USA,Behaviour 1998. of Dynamical Systems Near the Boundary of Stability; Nauka: Moscow, Russia, 1984. (In Russian) 12. Bautin, N. Behaviour of Dynamical Systems Near the Boundary of Stability; Nauka: Moscow, Russia, 1984. (In Russian) 13. Marsden,12. Bautin, J.; Cracken, N. Behaviour M. The of Dynamical Hopf Bifurcation Systems and Near Its Applicationsthe Boundary; Springer: of Stability New; Nauka: York, Moscow, NY, USA, Russia, 1976. 1984. (In Russian) 14. Homburg, A.J.; Sandstede, B. Homoclinic and heteroclinic bifurcations in vector fields. Handb. Dyn. Syst. 2010, 3, 379–524. 15. Glendinning, P.; Sparrow, C. Local and global behavior near homoclinic orbits. J. Statist. Phys. 1984, 34, 645–696. [CrossRef] 16. Guckenheimer, J.; Worfolk, P. Dynamical systems: Some computational problems. In Bifurcations and Periodic Orbits of Vector Fields; Schlomiuk, D., Ed.; Kluwer Academic Publishers: Berlin/Heidelberg, Germany, 1992; pp. 241–279. Mathematics 2021, 9, 352 16 of 17

17. Nikolov, S.; Nedkova, N. Dynamical behavior of a rigid body with one fixed point (gyroscope). Basic concepts and results. Open problems: A review. J. Appl. Comput. Mech. 2015, 1, 187–206. 18. Peixoto, M. Structural stability on two-dimensional manifolds. Topology 1962, 1, 101–120. [CrossRef] 19. Scott, S. Chemical Chaos; Clarendon Press: Oxford, UK, 1991. 20. Kuznetsov, Y. Elements of Applied Bifurcation Theory, 2nd ed.; Springer: New York, NY, USA, 1998. 21. Schuster, H.; Just, W. Deterministic Chaos: An Introduction; John Wiley & Sons: New York, NY, USA, 2006. 22. Phillipson, P.; Schuster, P. Analytics of bifurcation. Int. J. Bifurc. Chaos 1998, 8, 471–482. [CrossRef] 23. Nikolov, S. First Lyapunov value and bifurcation behavior of specific class of three-dimensional systems. Int. J. Bifurc. Chaos 2004, 14, 2811–2823. [CrossRef] 24. Shilnikov, L. A case of the existence of a denumerable set of periodic motions. Sov. Math. Dokl. 1965, 6, 163–166. (In Russian) 25. Shilnikov, L. A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium of saddle-focus type. Math. USSR-Sb. 1970, 10, 91–102. (In Russian) [CrossRef] 26. Tresser, C. About some theorems by Shilnikov, L.P. Ann. Inst. Henri Poincare Phys. Theor. 1984, 40, 441–461. 27. Glendinning, P. Subsidiary bifurcations near bifocal homoclinic orbits. Math. Camb. Philos. Soc. 1989, 105, 597–605. [CrossRef] 28. Fowler, A.; Sparrow, C. Bifocal homoclinic orbits in four dimensions. Nonlinearity 1991, 4, 1159–1182. [CrossRef] 29. Laing, C.; Glendinning, P. Bifocal homoclinic bifurcations. Phys. D Nonlinear Phenom. 1997, 102, 1–14. [CrossRef] 30. Lorenz, E. Deterministic non-periodic flow. J. Atmos. Sci. 1963, 20, 130–141. [CrossRef] 31. Rössler, O. An equation for continuous chaos. Phys. Lett. A 1976, 57, 397–398. [CrossRef] 32. Olsen, L.; Degn, H. Chaos in an enzyme reaction. Nature 1977, 267, 177–178. [CrossRef] 33. Olsen, L.; Degn, H. Oscillatory kinetics of the peroxidase-oxidase reaction in an open system. Experimental and theoretical studies. Biochim. Biophys. Acta 1978, 523, 321–334. [CrossRef] 34. Nikolov, S. Analysis of a Rossler type dynamical system. Mech. Transp. Commun. 2020, 18. in press. 35. Garcia, I.; Llibre, J.; Maza, S. Periodic orbits and their stability in Rössler prototype-4 system. Phys. Lett. A 2012, 376, 2234–2237. [CrossRef] 36. Sprott, J.; Linz, S. Algebraically simple chaotic flows. Int. J. Chaos Theory Appl. 2000, 5, 1–20. 37. Zhang, Z.; Fu, J. Hopf bifurcation and amplitude control of a Rössler prototype-4 system. Math. Pract. Theory 2016, 2016, 225–234. 38. Kuate, P.; Tchendjeu, A.; Fotsin, H. A modified Rössler prototype-4 system based on Chua’s diode nonlinearity: Dynamics, multistability, multiscroll generation and FPGA implementation. Chaos Solitons Fractals 2020, 140, 0213. [CrossRef] 39. Marzec, C.; Spiegel, E. Ordinary differential equations with strange attractors. SIAM J. Appl. Math. 1980, 38, 403–421. [CrossRef] 40. Nikolov, S.; Petrov, V. New results about route to chaos in Rossler system. Int. J. Bifurc. Chaos 2004, 14, 293–308. [CrossRef] 41. Nikolov, S.; Nedev, V. Bifurcation analysis and dynamic behaviour of an inverted pendulum with bounded control. J. Theor. Appl. Mech. 2016, 46, 17–32. [CrossRef] 42. Lichtenberg, A.; Lieberman, M. Regular and Chaotic Dynamics, 2nd ed.; Springer: New York, NY, USA, 1992. 43. Arnold, V.; Kozlov, V.; Neishtadt, A. Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed.; Springer: Berlin, Germany, 2006. 44. Llibre, J. Centers; their integrability and relations with the divergence. Appl. Math. Nonlinear Sci. 2016, 1, 79–86. [CrossRef] 45. Krupa, M. Robust heteroclinic cycles. J. Nonlinear Sci. 1997, 7, 129–176. [CrossRef] 46. Kuznecov, A.; Afraimovich, V. Heteroclinic cycles in the repressilator model. Chaos Solitons Fractals 2012, 45, 660–665. [CrossRef] 47. Shilnikov, L. The existence of a denumerable set of periodic motions in four-dimensional space in an extended neighborhood of a saddle-focus. Soviet Math. Dokl. 1967, 8, 54–58. (In Russian) 48. Shilnikov, A.; Shilnikov, L.; Barrio, R. Symbolic dynamics and spiral structures due to the saddle-focus bifurcations. In Chaos, CNN, Memristors and Beyond: A Festschrift for Leon Chua with DVD-ROM; Bilotta, E., Adamatzky, A., Chen, G., Eds.; World Scientific: London, UK, 2013; pp. 428–439. 49. Gonchenko, S.; Turaev, D.; Gaspard, P.; Nicolis, G. Complexity in the bifurcation structure of homoclinic loops to a saddle-focus. Nonlinearity 1997, 10, 409. [CrossRef] 50. Dufraine, E.; Danckaert, J. Some topological invariants for three-dimensional flows. Chaos 2001, 11, 443–448. [CrossRef] 51. Awrejcewicz, J. Numerical investigations of the constant and periodic motions of the human vocal cords including stability and bifurcation phenomena. Dyn. Stab. Syst. Int. J. 1990, 5, 11–28. [CrossRef] 52. Awrejcewicz, J.; Supel, B.; Lamarque, C.H.; Kudra, G.; Wasilewski, G.; Olejnik, P. Numerical and experimental study of regular and chaotic motion of triple physical pendulum. Int. J. Bifurc. Chaos 2008, 18, 2883–2915. [CrossRef] 53. Alligood, A.; Sauer, T.; Yorke, J. An Introduction to Dynamical Systems and Chaos; Springer: New York, NY, USA, 1996. 54. Fan, Y.; Chay, T. Crisis and topological entropy. Phys. Rev. E 1995, 51, 1012–1019. [CrossRef] 55. Sanjuan, M. Symmetry-restoring crises, period-adding and chaotic transitions in the cubic Van der Pol oscillator. J. Sound Vib. 1996, 193, 863–875. [CrossRef] 56. Uzunov, I.M.; Nikolov, S.G. Influence of the higher-order effects on the solutions of complex cubic-quintic Ginzburg–Landau equation. J. Mod. Opt. 2020, 67, 606–618. [CrossRef] 57. Gonchenko, S.; Ovsyannikov, I. On bifurcations of three-dimensional diffeomorphisms with a non-transversal heteroclinic cycle containing saddle-foci. Nonlinear Dyn. 2010, 6, 61–77. (In Russian) Mathematics 2021, 9, 352 17 of 17

58. Afraimovich, V.; Gonchenko, S.; Lerman, L.; Shilnikov, A.; Turaev, D. Scientiﬁc heritage of L.P. Shilnikov. Regul. Chaotic Dyn. 2014, 19, 435–460. [CrossRef] 59. Shilnikov, L. On a new type of bifurcation of multidimensional dynamical systems. Sov. Math. Dokl. 1969, 10, 1368–1371. (In Russian)