Mathematical and Computational Applications

Article Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid with Harmonic Excitation

Wojciech Smiechowicz´ 1, Théo Loup 1 and Paweł Olejnik 2,*

1 The International Faculty of Engineering, Lodz University of Technology, 36 Zwirki˙ Str., 90-001 Lodz, Poland; [email protected] (W.S.);´ [email protected] (T.L.) 2 Department of Automation, Biomechanics and Mechatronics, Faculty of Mechanical Engineering, Lodz University of Technology, 1/15 Stefanowskiego Str., 90-924 Lodz, Poland * Correspondence: [email protected]

 Received: 23 September 2019; Accepted: 15 October 2019; Published: 16 October 2019 

Abstract: This paper considers three dynamic systems composed of a mathematical pendulum suspended on a sliding body subjected to harmonic excitation. A comparative dynamic analysis of the studied parametric mutations of the rigid pendulum with inertial suspension point and damping was performed. The examined system with parametric mutations is solved numerically, where phase planes and Poincaré maps were used to observe the system response. Lyapunov exponents were computed in two ways to classify the dynamic behavior at relatively early stage of forced responses using two proven methods. The results show that with some parameters three systems exhibit a very similar dynamic behavior, i.e., quasi-periodic and even chaotic motions.

Keywords: parametric pendulum; Lyapunov exponents; quasi-periodic behavior; Chaos

1. Introduction Mechanical engineering or other fields always require elaboration based on numerical computations. Each mechanical system associated with periodic excitation behaves unpredictably. Excitations can come from the internal imperfections of the system, such as parametric asymmetry, eccentricity or external forces such as wind, road defects, etc. Any chaotic behavior is undesirable, and their consequences can lead to the destruction of the system. One method of qualitative determination of the state of the system is based on computations of Lyapunov exponents. Qualitatively, it informs about the form of the dynamic response of the system. These exponents are certain characteristic numerical values, creating a spectrum that helps in the qualitative assessment of the dynamics of the system. They determine the convergence or divergence of infinitely close trajectories that begin close to each other. To know the nature of system dynamics, all we need to know is the sign of the λ defined as follows:

1 δZ(t) λ = lim lim ln (1) t δZ0 0 t δZ0 →∞ → | | The Lyapunov exponent in Equation (1) is denoted by λ. Any positive value in the limit accounts for chaos in the system, while non-positive proves the regularity. There are many numerical methods of computing Lyapunov exponents, e.g., Wolf method, Rosenstein method, Kantz method, method based on neural network modification, synchronization method, and others, see [1–6]. There are two main approaches to numerical assessment in the literature: with known motion equations and a time series approach. For example, the first method of computing

Math. Comput. Appl. 2019, 24, 90; doi:10.3390/mca24040090 www.mdpi.com/journal/mca Math. Comput. Appl. 2019, 24, 90 2 of 15 Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 2 of 15

of computing the full spectrum of Lyapunov exponents is presented in [3], the calculation of the the full spectrum of Lyapunov exponents is presented in [3], the calculation of the largest Lyapunov largest Lyapunov exponent is given in [5,6], and the determination of the spectrum of Lyapunov exponent is given in [5,6], and the determination of the spectrum of Lyapunov exponents can be found exponents can be found in [7]. in [7]. From a practical point of view, Lyapunov exponents are used in various fields of science, such From a practical point of view, Lyapunov exponents are used in various fields of science, such as: as: rotor systems [8], electricity systems [9], aerodynamics [10]. rotor systems [8], electricity systems [9], aerodynamics [10].

2.2. The Parametric Pendulum and ItsIts MutationsMutations FigureFigure1 1 shows shows the the three three analyzed analyzed dynamical dynamical systems. systems. The The first first system system presented presented in in Figure Figure1a is1a a is a mathematical pendulum of the length attached to the moving slider of the point-focused mass mathematical pendulum of the length l0 attached to the moving slider of the point-focused mass M. The. The inertial inertial slider slider moves moves horizontally horizontally back back and and forth forth along along the thex -axisx-axis and and is is subjected subjected toto harmonicharmonic ( ) forcingforcingF (t)==F0 coscosω t, where, whereF0 denotes denotes amplitude amplitude of the of force,the force, and ω andstates thestates frequency the frequency of forcing. of Inforcing. the second In the system second depicted system depicted in Figure 1inb theFigure pendulum 1b the pendulum of the length of thel0 is length replaced byis replaced a parametric by a pendulum,parametric pendulum, the length ofthe which length is of described which is bydescribed the function by theS function(t) = l0 +(Asin) =(ω+()t), where A, whereis the amplitude is the amplitude of the periodic of the periodic change change of the lengthof the length about aboutl0. The generalized. The generalized coordinates coordinates of first of andfirst secondand second systems systems are: the are: angle the angleϕ between between the pendulum the pendulum and vertical and vertical axis and axis the and displacement the displacementx of the slider. of the In slider. the third In systemthe third shown system in Figureshown1 cin the Figure first rigid1c the pendulum first rigid of pendulum constant lengthof constant is replaced length by is anreplaced elastic by form, an elastic where form,k and wherec describe and linear describe stiffness linear and damping,stiffness and respectively. damping, respectively. The generalized The coordinatesgeneralized forcoordinates the third systemfor the can third be noted:system dynamicalcan be noted: elongation dynamical of the elongation , the of angle the spring,ϕ between the theangle pendulum between and the vertical pendulum axis and and the vertical displacement axis andx theof thedisplacement slider. of the slider.

(a) (b) (c)

FigureFigure 1.1.The The analyzed analyzed systems: systems: (a) rigid(a) rigid pendulum, pendulum, (b) first (b parametric) first parametric mutation, mutation, (c) second parametric(c) second mutationparametric in mutation an elastic in form. an elastic form

InIn orderorder to to simplify simplify mathematical mathematical modelling modelling of theof problemthe problem the following the following assumptions assumptions are made: are made: there is no friction in the inertial slider; •• there is no friction in the inertial slider; masses of the pendulum’s swinging body and slider are treated as point-focused masses; •• masses of the pendulum’s swinging body and slider are treated as point-focused masses; pendulum arm of a constant or variable length is massless. •• pendulum arm of a constant or variable length is massless. DynamicalDynamical analysis of the systemsystem mutations is preceded by derivation of equations of motion. ToTo findfind them,them, oneone cancan useuse NewtonNewton laws,laws, however,however, thisthis approachapproach requiresrequires aa betterbetter knowledgeknowledge ofof thethe system.system. A more universal and simplersimpler approachapproach basesbases onon derivingderiving themthem fromfrom thethe Euler–LagrangeEuler–Lagrange equationequation [[11].11]. 2.1. Mathematical Model of Rigid Pendulum 2.1. Mathematical Model of Rigid Pendulum The of the analyzed two-degrees-of-freedom (2 DoF) system is the sum of kinetic The kinetic energy of the analyzed two-degrees-of-freedom (2 DoF) system is the sum of kinetic energies of the slider (Ts) and the pendulum (Tb): energies of the slider () and the pendulum ():

11h . 2  . 2 . . . 2 2i =T = Ts++ Tb == [Mx ++(m x + +22xϕlcos cos+ϕ + ϕ l .)]. (2)(2) 22

Math. Comput. Appl. 2019, 24, 90 3 of 15

In the assumed configuration of the model the of the analyzed system is as follows:

U = mgl cos ϕ. (3) − Referring to Figure1, the following denotations have been used in Equations (2) and (3): M—mass . of the sliding body, m—mass of the mathematical pendulum, x—linear displacement and x— . of the sliding body, ϕ—angular displacement and ϕ—angular velocity of the pendulum, l—constant length of the rigid pendulum (Figure1a), g—gravitational constant. The Lagrangian L = T U satisfies the Euler–Lagrange equation for any of the assumed generalized − coordinates. Therefore, one writes: ! d ∂L ∂L . = Qi, where qi = [ϕ(t), x(t)], Qi = [0, F0 cos ωt]. (4) dt ∂qi − ∂qi

Substituting the Equations (2) and (3) for the kinetic and potential energy into the Lagrangian, respectively, it allows to derive the equation of motion for each of the generalized coordinates. The number of coupled equations of motion depends on the number of generalized coordinates. The equations of motion for the analyzed 2 DoF system are as follows:

for the generalized coordinate ϕ (pendulum angle): • .. .. x g ϕ + cos ϕ + sin ϕ = 0; (5) l l

for the generalized coordinate x (displacement of the slider): • .. .. . 2 (M + m)x + mlϕ cos ϕ mlϕ sin ϕ = F cos ωt. (6) − 0 The equations of motion (5)–(6) can be algebraically decoupled with respect to the second derivate:

. 2 .. mg sin ϕ cos ϕ+mlϕ sin ϕ+F cos ωt x = 0 , M+m sin2 ϕ (7) .. .. g ϕ = x cos ϕ sin ϕ. − l − l 2.2. Mathematical Model of First Mutation of the Rigid Pendulum The kinetic energy of the analyzed two-degree-of-freedom model with parametric pendulum is equal to the sum of the kinetic energy of the slider and kinetic energy of the pendulum body:   1 . 2 1  . . . 2  . . 2 T = Mx + m x + S(t) sin ϕ + S(t)ϕ cos ϕ + S(t) cos ϕ + S(t)ϕ sin ϕ . (8) 2 2 − The potential energy of the analyzed system is as follows:

U = mgS(t) cos ϕ. (9) − The Lagrangian satisfies the Euler–Lagrange equation for any of the assumed generalized coordinates. Therefore, one writes: ! d ∂L ∂L . = Qi, where qi = [ϕ(t), x(t)], Qi = [0, F0 cos ωt]. (10) dt ∂qi − ∂qi

From Equation (10) one can obtain two equations of motion:

for the generalized coordinate ϕ (pendulum angle): • ...... ϕS(t) + 2mS(t)ϕ + x cos ϕ + gsinϕ = 0; (11) Math. Comput. Appl. 2019, 24, 90 4 of 15

for the generalized coordinate x (displacement of the slider): • ..  . 2 ..   . . ..  (M + m)x mS(t)ϕ + S(t) sin ϕ cos ϕ 2S(t)ϕ + S(t)ϕ = F cos ωt. (12) − − 0 And after decoupling Equations (11) and (12) with respect to the second derivative:

..  . 2  . . .. m sin ϕ cos ϕ S(t) ϕ S(t) 2S(t)ϕ cos ϕF cos ωt sin ϕg(M+m) = − + − + − 0 − , ϕ S(t)(M+m m cos2 ϕ) S(t) S(t)(M+m m cos2 ϕ) − h . 2  .. i − (13) .. msinϕ S(t)ϕ + g cos ϕ S(t) +F0 cos ωt x = − . M+m m cos2 ϕ − 2.3. Mathematical Model of the Second Mutation of the Rigid Pendulum Similar derivation for the three-degree-of-freedom system was performed in [12]. The equation for kinetic energy was found follows:   . 2  . . . 2  . . 2 T = 1 Mx + 1 m x + s sin ϕ + sϕ cos ϕ + sϕ sin ϕ s cos ϕ 2 2 − (14) 1 . 2 1 h . 2 2 . 2 . . . i = 2 (M + m)x + 2 m s + s ϕ + 2x(s sin ϕ + sϕ cos ϕ) .

The potential energy of the analyzed system is equal to the potential energy of the pendulum body and potential energy of the spring:

1 U = k(s l )2 mgs cos ϕ. (15) 2 − 0 − The generalized form of the Euler–Lagrange equation for the third case is as follows: ! d ∂L ∂L ∂R . + . = Qi, where qi = [s(t), ϕ(t), x(t)], Qi = [0, 0, Fo cos ωt], (16) dt ∂qi − ∂qi ∂qi and R is Rayleigh dissipation function:

" #2 . 2 1 d(s l0 cs R = c − = . (17) 2 dt 2

The equations of motion have the following form:

for the generalized coordinate s (pendulum elongation): • .. .. .  . m s + x sin ϕ sϕ g cos ϕ + cs + k(s l ) = 0; (18) − − − 0 for the generalized coordinate ϕ (pendulum angle): • ...... sϕ + 2sϕ + x cos ϕ + g sin ϕ = 0; (19)

for the generalized coordinate x (displacement of the slider): • ..  .. . .  .. . 2 (M + m)x + m cos ϕ sϕ + 2sϕ + m sin ϕ s sϕ = F cos ωt. (20) − 0 Equations (18)–(20) can be algebraically decoupled with respect to the second derivate:

.. c . k . 2 .. s = s (s l0) + sϕ x sin ϕ + g cos ϕ, − m.. − m −. . .. − ϕ = 1 (2ϕs + x cos ϕ + g sin ϕ), (21) s . .. − (cs+k(s l )) sin ϕ+F cos ωt − 0 0 x = M . Math. Comput. Appl. 2019, 24, 90 5 of 15

3. Methods To numerically solve the systems of second order ordinary differential Equations (7), (13) and (21), they were written in the form of first order differential equations. The vector of initial conditions h . . i for the first and second system is assumed, ϕ, ϕ, x, x = [0, 0, 0, 0]. In the case of the third system of three degrees of freedom, the additional equation associated with the pendulum elongation only represents the incremental elongation of the spring measured from its free length l0. The vector of h . . . i initial conditions is assumed as well, s, s, ϕ, ϕ, x, x = [0, 0, 0, 0, 0, 0]. Numerical analysis of the presented models was carried out in two different environments. Phase planes and Poincaré maps were found using a program prepared in the Python programming language, and computations of Lyapunov exponents were carried out using codes written in Matlab environment. ODEINT from the SciPy Python library and ODE45 from the Matlab library was used to solve the system of ordinary differential equations.

Computation of Lyapunov Exponents Lyapunov exponents are one of the most commonly used tools in analyzing the impact of small disturbances on system solution. These are the values that determine the exponential convergence or divergence of trajectories beginning close to each other [13]. Two methods of computing Lyapunov exponents were used in the study. The first method is described in [14]. This method simultaneously solves the first-order ODE system, its variational equations and determines the spectrum of Lyapunov exponents. The second method used for computations is available on the Mathworks file sharing page [15]. The numerical code is based on the algorithm for computing Lyapunov exponents on the basis of ODE proposed in the work [4]. Both selected methods were validated for the written in Equation (22) with the parameters σ = 10.0, R = 28, and B = 8/3 and also initial conditions [0.0, 1.0, 0.0]. Table1 shows a comparison of obtained Lyapunov exponents spectrum with a reference values found in [16]:

. . . X = σ(Y X),Y = X(R Z) Y,Z = XY BZ. (22) − − − −

Table 1. Comparison of Lyapunov exponents (time is measured in seconds).

tend = 10, 000 tend = 100, 000 Exponents Literature Method 1 Method 2 Literature Method 1 Method 2

λ1 0.9022 0.903269 0.9012 0.9051 0.90609 0.903240 λ2 0.0003 0.0003 0.00209 0.0000 0.00003 0.001804 λ 14.5691 14.5703 14.566 14.5718 14.5728 14.5681 3 − − − − − −

Computed spectra of Lyapunov exponents are very similar to the reference values taken from the literature [13,14], respectively.

4. Results Presented systems were examined for the set of parameters presented in Table2. Frequency of harmonic forcing ω was assumed in the range from 1.24 to 9.24 rad/s. Vector of state h . . i h . . i variables for the first and second system is assumed, y = y1, y1, y2, y2 = ϕ, ϕ, x, x , and for the third . . . h . . . i system is taken as follows, y = [y1, y1, y2, y2, y3, y3] = s, s, ϕ, ϕ, x, x . Math. Comput. Appl. 2019, 24, 90 6 of 15

Table 2. Parameters of the simulations (values of M, m, F0 and l0 are common for all the systems).

First Parametric Second Parametric Parameter Rigid Pendulum Mutation Mutation M [kg] 5 m [kg] 0.3 F0 [N] 4 Math. Comput. Appl.l0 [m] 2019, 24, x FOR PEER REVIEW 0.35 6 of 15 Math. Comput. Appl.A [m] 2019, 24, x FOR PEER REVIEW 0.15; 0.1; 0.001 6 of 15 − − Frequencyc [Ns of/m] harmonic forcing was assumed in the range from 1.24 0.05;to 9.24 9.05 rad/s. Vector of − − k [N/m] 40; 2000 stateFrequency variables for of harmonicthe first and forcing second− system was assumed is assumed, in − the = range[ from, , 1.24, ] =[to 9.24, ,, rad/s.], and Vector for the of state variables for the first andFigures second=[ 2–5 system, , Figuresis, assumed, , 6–9, ] and =[, 15–17[ , ,, ,,,Figures,] ] =[ 10–14, and ,, 18–20], and for the third systemResults is taken as follows, = . Table 3 Tables 4 and 6 Tables 5 and 7 third system is taken as follows, =[, ,, ,, ] = [, , , ,,]. 4.1. Numerical Simulation of the Rigid Pendulum 4.1. Numerical Numerical Simulation Simulation of of the the Rigid Pendulum Figures 2–5 show phase planes and Poincaré maps for different frequencies of harmonic excitationFigures 2 .– 2–5By5 show theshow assumed phase phase planes definition,planes and Poincarand th Poincarée mapé maps definition maps for di ffforerent results different frequencies in stroboscopic frequencies of harmonic analysis of harmonic excitation of the excitationωphase. By thespace assumed. atBy time the definition, instancesassumed thebeingdefinition, map a multiple definition the map of results thedefinition period in stroboscopic results=2/ in .stroboscopic analysisThe map of presentation theanalysis phase of space on the a phaseatshaded time space instancesbackground at time being instancescreated a multiple by being a cross-section ofa multiple the period of ththeTe= phaseperiod2π/ ωspace .=2/ The has map most. The presentation likely map presentationbeen onfirst a shadedused on ina shadedbackgroundthe work background [17]. created The corresponding bycreated a cross-section by a cross-section values of the ofphase Lyapunov of spacethe phase exponents has most space likely are has shown beenmost first likelyin Table used been in3. the first work used [17 in]. theThe work corresponding [17]. The corresponding values of Lyapunov values exponents of Lyapunov are shownexponents in Table are shown3. in Table 3.

Figure 2. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 3.64. FigureFigure 2. 2. PhasePhase planes planes (gray (gray lines) lines) and and Poinca Poincarréé mapsmaps (red (red and and green green dots) dots) for for =ω = 3.64.3.64.

FigureFigure 3. 3. PhasePhase planes planes (gray (gray lines) lines) and and Poinca Poincarréé mapsmaps (red (red and and green green dots) dots) for for ω = =4.94. 4.94. Figure 3. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 4.94.

Figure 4. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 7.24. Figure 4. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 7.24.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 6 of 15

Frequency of harmonic forcing was assumed in the range from 1.24 to 9.24 rad/s. Vector of state variables for the first and second system is assumed, =[, ,, ] =[, ,,], and for the third system is taken as follows, =[, ,, ,, ] = [, , , ,,].

4.1. Numerical Simulation of the Rigid Pendulum Figures 2–5 show phase planes and Poincaré maps for different frequencies of harmonic excitation . By the assumed definition, the map definition results in stroboscopic analysis of the at time instances being a multiple of the period =2/. The map presentation on a shaded background created by a cross-section of the phase space has most likely been first used in the work [17]. The corresponding values of Lyapunov exponents are shown in Table 3.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 7 of 15

As shown in the Poincaré maps in Figures 2–4, a system with a rigid pendulum of constant Figure 2. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 3.64. length tends to be quasi-periodic. For the tested ∈ [1.24, 9.24] rad/s with a step of 0.1, the system is stable and does not show chaotic behavior (in a that is far from a resonance frequency of excitation). However, forms of quasi-periodic behavior change with frequency, but do not stabilize on periodic orbits. The computed spectra of Lyapunov exponents indicate that the behavior of the

system is two-periodic. Lyapunov exponents and tend to converge to zero, while the others are negative. Both methods provide slightly different but similar results. Interesting trajectories of the convergence of Lyapunov exponents are presented in Figure 5.

Table 3. Spectra of Lyapunov exponents for the system with rigid pendulum.

Parameter Method Method 150 0.01777 0.01497 −0.00204 −0.03070 0.02563 0.02160 −0.00294 −0.04429 Math. Comput. Appl. 2019, 24, 90 7 of 15 3.64 Figure350 3. 0.00884Phase planes 0.00834 (gray lines) −0.00117 and Poinca −0.01601ré maps (red 0.01275 and green 0.01203 dots) for −0.00169 = 4.94 . −0.02310 650 0.00621 0.00571 −0.00202 −0.00991 0.00896 0.00824 −0.00291 −0.01430 150 0.03443 0.01742 −0.01842 −0.03344 0.04968 0.02513 −0.02657 −0.04824 4.94 350 0.01399 0.00900 −0.00612 −0.01698 0.02019 0.01299 −0.00883 −0.02450 650 0.01221 0.00626 −0.00852 −0.00998 0.01761 0.00904 −0.01229 −0.01439 150 0.01524 0.00940 −0.00030 −0.02436 0.02199 0.01357 −0.00043 −0.03514 7.24 350 0.00896 0.00689 −0.00060 −0.01525 0.01293 0.00994 −0.00087 −0.02201 650 0.00578 0.00517 −0.00117 −0.00978 0.00834 0.00745 −0.00168 −0.01411

The and exponents stabilize, while and oscillate with decreasing amplitude, which was necessary to prove. The behavior of a rigid pendulum with a fixed length can be compared with the behavior of the third system with a flexible pendulum with assumed damping constant. Figure 4. Phase planes (gray lines) and Poincaré maps (red and green dots) for ω = 7.24. Figure 4. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 7.24.

FigureFigure 5.5. TimeTime historyhistory ofof LyapunovLyapunov exponentsexponents forforω = =4.94 4.94 (the (the case case of rigid of rigid pendulum) pendulum) Table 3. Spectra of Lyapunov exponents for the system with rigid pendulum. 4.2. Numerical Simulation of the First Mutation of the Rigid Pendulum Parameter Method 1 Method 2 Selected behaviors of the dynamical model of two degrees of freedom with a parametric forcing areω presentedtend in Figuresλ1 5–7. Tableλ2 4 showsλ3 the computedλ4 spectraλ1 of Lyapunovλ2 exponents.λ3 λ4 150 0.01777 0.01497 0.00204 0.03070 0.02563 0.02160 0.00294 0.04429 − − − − 3.64 350 0.00884 0.00834 0.00117 0.01601 0.01275 0.01203 0.00169 0.02310 − − − − 650 0.00621 0.00571 0.00202 0.00991 0.00896 0.00824 0.00291 0.01430 − − − − 150 0.03443 0.01742 0.01842 0.03344 0.04968 0.02513 0.02657 0.04824 − − − − 4.94 350 0.01399 0.00900 0.00612 0.01698 0.02019 0.01299 0.00883 0.02450 − − − − 650 0.01221 0.00626 0.00852 0.00998 0.01761 0.00904 0.01229 0.01439 − − − − 150 0.01524 0.00940 0.00030 0.02436 0.02199 0.01357 0.00043 0.03514 − − − − 7.24 350 0.00896 0.00689 0.00060 0.01525 0.01293 0.00994 0.00087 0.02201 − − − − 650 0.00578 0.00517 0.00117 0.00978 0.00834 0.00745 0.00168 0.01411 − − − −

As shown in the Poincaré maps in Figures2–4, a system with a rigid pendulum of constant length tends to be quasi-periodic. For the tested ω [1.24, 9.24] rad/s with a step of 0.1, the system is stable ∈ and does not show chaotic behavior (in a vibration that is far from a resonance frequency of excitation). However, forms of quasi-periodic behavior change with frequency, but do not stabilize on periodic orbits. The computed spectra of Lyapunov exponents indicate that the behavior of the system is two-periodic. Lyapunov exponents λ1 and λ2 tend to converge to zero, while the others are negative. Both methods provide slightly different but similar results. Interesting trajectories of the convergence of Lyapunov exponents are presented in Figure5. Math. Comput. Appl. 2019, 24, 90 8 of 15

The λ2 and λ4 exponents stabilize, while λ1 and λ2 oscillate with decreasing amplitude, which was necessary to prove. The behavior of a rigid pendulum with a fixed length can be compared with the behavior of the third system with a flexible pendulum with assumed damping constant.

4.2. Numerical Simulation of the First Mutation of the Rigid Pendulum

Math. Comput.Selected Appl. behaviors 2019, 24, x ofFOR the PEER dynamical REVIEW model of two degrees of freedom with a parametric forcing 8 of 15 areMath. presented Comput. Appl. in 2019 Figures, 24, x5 FOR–7. TablePEER 4REVIEW shows the computed spectra of Lyapunov exponents. 8 of 15

Figure 6. Phase planes (gray lines) and Poincaré maps (red and green dots) for =0.1, = 4.94. Figure 6. Phase planes (gray lines) and Poincaré maps (red and green dots) for A = 0.1, ω = 4.94. Figure 6. Phase planes (gray lines) and Poincaré maps (red and green dots) for =0.1, = 4.94.

FigureFigure 7. 7.PhasePhase planes planes (gray (gray lines) lines) and and Poinca Poincarré émapsmaps (red (red and and green green dots) dots) for for =0.1A = 0.1,, ω =7.64. 7.64. Figure 7.Table Phase 4. planesSpectra (gray of Lyapunov lines) and exponents Poincaré maps for the (red first and mutation green dots) of rigid forpendulum. =0.1, = 7.64. Replacing a rigid pendulum with a fixed length by a parametric pendulum makes the system moreParameters Replacingunpredictable. a rigid pendulum Methodwith a fixed 1 length by a parametric pendulum Method makes 2 the system moreA unpredictable. tend λ1 λ2 λ3 λ4 λ1 λ2 λ3 λ4 ω Table 4. Spectra of Lyapunov exponents for the first mutation of rigid pendulum. 150Table 4. 1.05622 Spectra of 0.00771Lyapunov exponents0.03341 for1.02892 the first1.51680 mutation of 0.01113 rigid pendulum.0.04823 1.47763 0.1 − − − − Parameters 350 0.75607 0.00573Method 1 0.01674 0.74642 1.52133 0.00826 Method 0.024172 1.50741 Parameters 3.64 Method 1− − Method− 2 − 650 0.69003 0.00404 0.01001 0.68438 1.53322 0.00582 0.01440 1.52512 − − − − 150 0.01552 0.00254 0.00010 0.02015 0.02239 0.00366 0.00015 0.02908 0.1 150 1.05622 0.00771 −0.03341− −1.02892− 1.51680 0.01113 −0.04823− −1.47763− 0.1 350 0.00809 0.00229 0.00002 0.00951 0.01167 0.00331 0.00003 0.01372 4.94 350150 0.75607 1.05622 0.00573 0.00771 −0.01674 −0.03341− −0.74642 −1.02892− 1.52133 1.51680 0.00826 0.01113 −0.02417 −0.04823− −1.50741−1.47763− 3.640.1 650 0.00589 0.00330 0.00043 0.00936 0.00850 0.00476 0.00062 0.01351 350 0.75607 0.00573 −0.01674− −0.74642− 1.52133 0.00826 −0.02417− −1.50741− 3.64 650 150 0.69003 0.99155 0.00404 0.00947 −0.010010.03341 −0.684380.97197 1.533221.76220 0.00582 0.01367 −0.014400.04821 −1.525121.73450 0.15 − − − − 150650 350 0.01552 0.69003 1.28874 0.00254 0.00404 0.00648 −0.00010 −0.010010.01674 −0.02015 −0.684381.27534 0.02239 1.533221.44346 0.00366 0.00582 0.00935 −0.00015 −0.014400.02415 −0.02908−1.525121.42404 0.1 7.24 − − − − 350150 650 0.00809 0.01552 1.24185 0.00229 0.00254 0.00444 −0.00002 −0.000100.00997 −0.00951 −0.020151.23712 0.01167 0.022391.28666 0.00331 0.00366 0.00641 −0.00003 −0.000150.01439 −0.01372−0.029081.27978 4.940.1 − − − − 350 0.00809 0.00229 −0.00002 −0.00951 0.01167 0.00331 −0.00003 −0.01372 4.94 650 0.00589 0.00330 −0.00043 −0.00936 0.00850 0.00476 −0.00062 −0.01351 Replacing150650 0.99155 a0.00589 rigid pendulum 0.00947 0.00330 with−0.03341 −0.00043 a fixed −0.97197 −0.00936 length by 1.76220 a0.00850 parametric 0.01367 0.00476 pendulum −0.04821 −0.00062 makes −1.73450 the−0.01351 system 0.15 more unpredictable.350150 1.28874 0.99155 0.00648 0.00947 −0.01674 −0.03341 −1.27534 −0.97197 1.44346 1.76220 0.00935 0.01367 −0.02415 −0.04821 −1.42404−1.73450 7.240.15 350 1.28874 0.00648 −0.01674 −1.27534 1.44346 0.00935 −0.02415 −1.42404 7.24 650 1.24185 0.00444 −0.00997 −1.23712 1.28666 0.00641 −0.01439 −1.27978 650 1.24185 0.00444 −0.00997 −1.23712 1.28666 0.00641 −0.01439 −1.27978 The system is not able to achieve periodic behavior for the frequencies tested. Figure 6 shows chaoticThe or systemquasi-periodic is not able behavior, to achieve but the periodic definitely beha positivevior for value the frequencies of Lyapunov tested. exponents Figure confirms 6 shows thechaotic chaotic or quasi-periodic behavior of the behavior, system. but Figure the definitely8 presents positive the time value history of Lyapunov of the Lyapunov exponents exponents confirms spectrum.the chaotic You behavior can see of that the the system. exponents Figure are 8 stablepresents and the the time system history is chaotic. of the TheLyapunov system exponents may also exhibitspectrum. quasi-periodic You can see behavior, that the exponentsas shown inare Figure stable 7. and Amplitude the system is of chaotic.periodic The change system in length may also of aboutexhibit quasi-periodic has a significant behavior, impact as on shown the behavior in Figure of 7.the Amplitude system. When of the periodic value change varies in fromlength 0.1 of about has a significant impact on the behavior of the system. When the value varies from 0.1

Math. Comput. Appl. 2019, 24, 90 9 of 15

The system is not able to achieve periodic behavior for the frequencies tested. Figure6 shows chaotic or quasi-periodic behavior, but the definitely positive value of Lyapunov exponents confirms the chaotic behavior of the system. Figure8 presents the time history of the Lyapunov exponents spectrum. You can see that the exponents are stable and the system is chaotic. The system may also exhibitMath. Comput. quasi-periodic Appl. 2019, 24 behavior,, x FOR PEER as REVIEW shown in Figure7. Amplitude A of periodic change in length 9 of 15 of about l has a significant impact on the behavior of the system. When the A value varies from to 0.15, 0and the harmonic forcing frequency increases by about 0.1, the system exhibits chaotic 0.1 to 0.15Math., andComput. the Appl. harmonic 2019, 24, x FOR forcing PEER REVIEW frequency increases by about 0.1, the system exhibits 9 of 15 chaotic behavior, as shown in Figure 9. Such behavior can easily be seen on the Poincaré map and this is behavior, as shown in Figure9. Such behavior can easily be seen on the Poincar é map and this is clearlyto indicated 0.15, and theby harmonicdefinitely forcing positive frequency Lyapunov increases exponent. by about 0.1By, thechanging system exhibitsparameter chaotic , quasi- clearly indicated by definitely positive Lyapunov exponent. By changing parameter A, quasi-periodic periodicbehavior, or chaotic as shown behavior in Figure is observed 9. Such behavior in different can easily frequency be seen ranges.on the Poincaré map and this is or chaoticclearly behavior indicated is observed by definitely in di positivefferent frequencyLyapunov exponent. ranges. By changing parameter , quasi- periodic or chaotic behavior is observed in different frequency ranges.

= 4.94 FigureFigure 8. Time 8. Time history history ofof LyapunovLyapunov exponents exponents for for ω = .4.94. Figure 8. Time history of Lyapunov exponents for = 4.94.

FigureFigure 9. Phase 9. Phase planes planes (gray (gray lines) lines) and and Poincar Poincaréé mapsmaps (red (red and and green green dots) dots) for for = 0.15A =, 0.15, = 5.04ω .= 5.04.

4.3. Numerical4.3. Numerical Simulation Simulation of the of the Second Second Mutation Mutation ofof Rigid Pendulum Pendulum InFigure FiguresIn 9. Figures Phase10–12 10–12 planesphase phase (gray planes planes lines) and and and Poincar Poincaré Poincaé rémaps maps maps for (red the the andsecond second green mutation mutation dots) for are presented. are = presented.0.15, Table = 5.04 Table. 5 presents5 thepresents computed the computed spectra spectra of Lyapunov of Lyapunov exponents. exponents. 4.3. InNumerical the case Simulation of a system of with the Se acond flexible Mutation pendulum, of Rigid it isPendulum worth noting the effect of damping. In the case ofIn weak Figures damping 10–12 phase and lowplanes frequency and Poincaré of harmonic maps for forcing, the second the system mutation shows are presented. quasi-periodic Table behavior,5 presents as the shown computed in Figure spectra 10. of Lyapunov exponents. Figure 11 shows that at higher forcing frequencies the system remains chaotic. Strong damping makes the system stiffer, and some weak damping gives the system more freedom. This hypothesis was verified in Figure 12, on which periodic movement is clearly visible. Numerical analysis showed that for ω [3.14, 9.24], where C = 9.05, behaviors are periodic. The computed values of Lyapunov exponents ∈ (see Table5) confirm the observed dynamic behavior. Figure 13 proves that for quasi-periodic motion, the spectrum of Lyapunov exponents changes with an increasingly smaller amplitude. Figure 14 shows that in the caseFigure of 10. periodic Phase planes conservation (gray lines) and and Poincaré high energy maps (red, dispersion green and blue in the dots) system for = spectra 0.05, of Lyapunov exponents are=40 more and stable = 2.84 than. in the case of quasi-periodicity.

Figure 10. Phase planes (gray lines) and Poincaré maps (red, green and blue dots) for = 0.05, =40 and = 2.84.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 9 of 15 to 0.15, and the harmonic forcing frequency increases by about 0.1, the system exhibits chaotic behavior, as shown in Figure 9. Such behavior can easily be seen on the Poincaré map and this is clearly indicated by definitely positive Lyapunov exponent. By changing parameter , quasi- periodic or chaotic behavior is observed in different frequency ranges.

Figure 8. Time history of Lyapunov exponents for = 4.94.

Figure 9. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 0.15, = 5.04.

4.3. Numerical Simulation of the Second Mutation of Rigid Pendulum In Figures 10–12 phase planes and Poincaré maps for the second mutation are presented. Table Math. Comput. Appl. 2019, 24, 90 10 of 15 5 presents the computed spectra of Lyapunov exponents.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 10 of 15 Figure 10. Phase planes (gray lines) and Poincaré maps (red, green and blue dots) for C = 0.05, k = 40 Math. Comput.Figure 10.Appl. Phase 2019, planes24, x FOR (gray PEER lines) REVIEW and Poincaré maps (red, green and blue dots) for = 0.05, 10 of 15 and ω = 2.84. =40 and = 2.84.

Figure 11. Phase planes and Poincaré maps (colors as above) for = 0.05, =40 and = 5.84. FigureFigure 11. 11. PhasePhase planes planes and and Poincaré Poincar émapsmaps (colors (colors as as above) above) for for C == 0.050.05,, =40k = 40 andand ω= =5.84. 5.84. Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 11 of 15

Figure 11 shows that at higher forcing frequencies the system remains chaotic. Strong damping makes the system stiffer, and some weak damping gives the system more freedom. This hypothesis was verified in Figure 12, on which periodic movement is clearly visible. Numerical analysis showed that for ∈ [3.14, 9.24], where = 9.05, behaviors are periodic. The computed values of Lyapunov exponents (see Table 5) confirm the observed dynamic behavior. Figure 13 proves that for quasi- periodic motion, the spectrum of Lyapunov exponents changes with an increasingly smaller amplitude. Figure 14 shows that in the case of periodic conservation and high energy dispersion in the system spectra of Lyapunov exponents are more stable than in the case of quasi-periodicity. Figure 12. Phase planes and Poincaré maps (colors as above) for C = 9.05, k = 40 and ω = 3.14. Figure 12. Phase planes and Poincaré maps (colors as above) for = 9.05, =40 and = 3.14. Figure 12. Phase planes and Poincaré maps (colors as above) for = 9.05, =40 and = 3.14. Table 5. Spectra of Lyapunov exponents for the system with second mutation of rigid pendulum. Table 5. Spectra of Lyapunov exponents for the system with second mutation of rigid pendulum. Parameters Method Parameters Method 150 0.01389 0.00730 −0.02455 −0.03430 −0.05477 −0.07696 = 0.05 150350 0.01389 0.00888 0.00730 0.00554 −0.02455 −0.01581 −0.03430 −0.02795 −0.05477 −0.05893 −0.07696 −0.07962 = 0.05 = 2.84 350 0.00888 0.00554 −0.01581 −0.02795 −0.05893 −0.07962 = 2.84 650 0.00393 0.00330 −0.00826 −0.01600 −0.06848 −0.08189 650150 0.00393 0.03793 0.00330 0.00730 −0.00826 −0.03099 −0.01600 −0.03481 −0.06848 −0.05579 −0.08189 −0.09234 = 0.05 150350 0.03793 0.03348 0.00730 0.00554 −0.03099 −0.00579 −0.03481 −0.02795 −0.05579 −0.06891 −0.09234 −0.10467 = 0.05 = 5.84 350 0.03348 0.00554 −0.00579 −0.02795 −0.06891 −0.10467 = 5.84 650 0.02850 0.00393 −0.00646 −0.01600 −0.07304 −0.10459 650150 0.02850 0.00730 0.00393 −0.01519 −0.00646 −0.04510 −0.01600 −0.05959 −0.07304 −5.33830 −0.10459 −24.73932 = 9.05 150350 0.00730 0.00554 −0.01519 −0.00924 −0.04510 −0.02146 −0.05959 −0.02795 −5.33830 −5.36490 −24.73932 −24.76804 = 9.05 = 3.14 350 0.00554 −0.00924 −0.02146 −0.02795 −5.36490 −24.76804 = 3.14 650 0.00393 −0.00528 −0.01458 −0.01600 −5.37365 −24.77849 650 0.00393 −0.00528 −0.01458 −0.01600 −5.37365 −24.77849 Figure 13. Time history of Lyapunov exponents starting at t = 50 for C = 0.05, k = 40, ω = 2.84. ParametersFigure 13. Time history of Lyapunov exponents startingMethod at =50 for =0.05, =40, = 2.84. Parameters Method 150 0.01989 0.01054 −0.03525 −0.04980 −0.07995 −0.10981 = 0.05 150350 0.01989 0.01282 0.01054 0.00799 −0.03525 −0.02280 −0.04980 −0.04032 −0.07995 −0.08502 −0.10981 −0.11487 = 0.05 = 2.84 350 0.01282 0.00799 −0.02280 −0.04032 −0.08502 −0.11487 = 2.84 650 0.00568 0.00476 −0.01192 −0.02308 −0.09879 −0.11814 650150 0.00568 0.05473 0.00476 0.01054 −0.01192 −0.04471 −0.02308 −0.05022 −0.09879 −0.08049 −0.11814 −0.13322 = 0.05 150350 0.05473 0.04830 0.01054 0.00799 −0.04471 −0.00836 −0.05022 −0.04033 −0.08049 −0.09942 −0.13322 −0.15100 = 0.05 = 5.84 350 0.04830 0.00799 −0.00836 −0.04033 −0.09942 −0.15100 = 5.84 650 0.04112 0.00568 −0.00932 −0.02308 −0.10537 −0.15089 650150 0.04112 0.01054 0.00568 −0.02192 −0.00932 −0.06507 −0.02308 −0.08598 −0.10537 −7.70154 −0.15089 −35.69129 = 9.05 150350 0.01054 0.00799 −0.02192 −0.01333 −0.06507 −0.03096 −0.08598 −0.04032 −7.70154 −7.73992 −35.69129 −35.73272 = 9.05 = 3.14 350 0.00799 −0.01333 −0.03096 −0.04032 −7.73992 −35.73272 = 3.14 650 0.00568 −0.00762 −0.02103 −0.02308 −7.75253 −35.74780 650 0.00568 −0.00762 −0.02103 −0.02308 −7.75253 −35.74780 In the case of a system with a flexible pendulum, it is worth noting the effect of damping. In the caseIn of the weak case ofdamping a system and with low a flexible frequency pendulum, of harmonic it is worth forcing, noting the thesystem effect shows of damping. quasi-periodic In the casebehavior, of weak as dampingshown in Figureand low 10. frequency of harmonic forcing, the system shows quasi-periodic behavior, as shown in Figure 10.

Figure 14. Time history of Lyapunov exponents starting at =50 for =9.05, k = 40 and = 3.14.

4.4. Observation of Similarities in the Behaviour of Systems with a Rigid Pendulum and Its Mutations In this section, we present the results for a special case when all the analyzed systems behave in a similar way. In the case of the second system, the amplitude of the pendulum length is A = 0.001, while the parameters of the third system are as follows, k = 2000 and C = 0.05. The previously presented spectra of Lyapunov exponents have proved that the most accurate

results are obtained for a longer observation time, therefore in this part only the results for time = 650 are presented. Figures 15–17 show the behavior of the second system, while the spectrum of Lyapunov exponents can be found in Table 6.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 11 of 15 Math. Comput. Appl. 2019, 24, 90 11 of 15 Figure 11 shows that at higher forcing frequencies the system remains chaotic. Strong damping makes the system stiffer, and some weak damping gives the system more freedom. This hypothesis Table 5. Spectra of Lyapunov exponents for the system with second mutation of rigid pendulum. was verified in Figure 12, on which periodic movement is clearly visible. Numerical analysis showed that forParameters ∈ [3.14, 9.24], where = 9.05, behaviors are periodic. Method 1 The computed values of Lyapunov exponents (see Table 5) confirm the observed dynamic behavior. Figure 13 proves that for quasi- tend λ1 λ2 λ3 λ4 λ5 λ6 periodic motion, the spectrum of Lyapunov exponents changes with an increasingly smaller 150 0.01389 0.00730 0.02455 0.03430 0.05477 0.07696 C = 0.05 − − − − amplitude. Figure350 14 shows 0.00888 that in the 0.00554 case of periodic0.01581 conservation0.02795 and high0.05893 energy dispersion0.07962 in ω = 2.84 − − − − the system spectra650 of Lyapunov 0.00393 exponents 0.00330 are more0.00826 stable than 0.01600in the case of0.06848 quasi-periodicity.0.08189 − − − − 150 0.03793 0.00730 0.03099 0.03481 0.05579 0.09234 C = 0.05 − − − − 350 0.03348 0.00554 0.00579 0.02795 0.06891 0.10467 ω = 5.84 − − − − 650 0.02850 0.00393 0.00646 0.01600 0.07304 0.10459 − − − − 150 0.00730 0.01519 0.04510 0.05959 5.33830 24.73932 C = 9.05 − − − − − 350 0.00554 0.00924 0.02146 0.02795 5.36490 24.76804 ω = 3.14 − − − − − 650 0.00393 0.00528 0.01458 0.01600 5.37365 24.77849 − − − − − Parameters Method 2

tend λ1 λ2 λ3 λ4 λ5 λ6 150 0.01989 0.01054 0.03525 0.04980 0.07995 0.10981 C = 0.05 − − − − 350 0.01282 0.00799 0.02280 0.04032 0.08502 0.11487 ω = 2.84 − − − − 650 0.00568 0.00476 0.01192 0.02308 0.09879 0.11814 − − − − 150 0.05473 0.01054 0.04471 0.05022 0.08049 0.13322 C = 0.05 − − − − 350 0.04830 0.00799 0.00836 0.04033 0.09942 0.15100 ω = 5.84 − − − − 650 0.04112 0.00568 0.00932 0.02308 0.10537 0.15089 − − − − 150 0.01054 0.02192 0.06507 0.08598 7.70154 35.69129 C = 9.05 − − − − − 350 0.00799 0.01333 0.03096 0.04032 7.73992 35.73272 ω = 3.14 − − − − − Figure 13. Time650 history 0.00568 of Lyapunov 0.00762exponents starting0.02103 at =500.02308 for =0.057.75253, =40, =35.74780 2.84. − − − − −

Figure 14. Time history of Lyapunov exponents starting at t = 50 for C = 9.05, k = 40 and ω = 3.14. Figure 14. Time history of Lyapunov exponents starting at =50 for =9.05, k = 40 and = 3.14. 4.4. Observation of Similarities in the Behaviour of Systems with a Rigid Pendulum and Its Mutations 4.4. Observation of Similarities in the Behaviour of Systems with a Rigid Pendulum and Its Mutations In this section, we present the results for a special case when all the analyzed systems behave in a similarIn this way. section, In the we case present of the the second results system, for a spec theial amplitude case when of theall the pendulum analyzed length systems is A behave= 0.001, in whilea similar the parametersway. In the ofcase the of third the second system aresyst asem, follows, the amplitudek = 2000 of and theC pendulum= 0.05. length is A = 0.001, whileThe the previously parameters presented of the third spectra system of are Lyapunov as follows, exponents k = 2000have and C proved = 0.05. that the most accurate The previously presented spectra of Lyapunov exponents have proved that the most accurate results are obtained for a longer observation time, therefore in this part only the results for time t0 = 650 areresults presented. are obtained for a longer observation time, therefore in this part only the results for time = 650 Figuresare presented. 15–17 show the behavior of the second system, while the spectrum of Lyapunov exponents can beFigures found in15–17 Table show6. the behavior of the second system, while the spectrum of Lyapunov exponents can be found in Table 6.

Math. Comput. Appl. 2019, 24, 90 12 of 15 Math. Comput. Appl. 2019,, 24,, xx FORFOR PEERPEER REVIEWREVIEW 12 12 of of 15 15 Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 12 of 15

Figure 15. Phase planes (gray lines) and Poincaré maps (red and green dots) for = 0.001, =3.64. Figure 15. PhasePhase planes planes (gray (gray lines) lines) and and Poinca Poincarréé mapsmaps (red (red and and green green dots) dots) for for A == 0.0010.001,,, ω=3.64= 3.64...

FigureFigure 16. 16. PhasePhase planes planes and and Poincaré Poincaré mapsmaps (colors (colors as as above) above) for for =0.001A = 0.001,, ω= =4.94. 4.94. Figure 16. PhasePhase planesplanes andand PoincaréPoincaré maps (colors as above) for =0.001,, = 4.94..

FigureFigure 17. 17. PhasePhase planes planes and and Poincaré Poincaré mapsmaps for for =0.001A = 0.001,, ω= =7.24. 7.24. Figure 17. PhasePhase planesplanes andand PoincaréPoincaré mapsmaps forfor =0.001,, = 7.24.. Table 6. Spectra of Lyapunov exponents for the second mutation of the system with rigid pendulum. Table 6. Spectra of Lyapunov exponents for the second mutation of the system with rigid pendulum. Table 6. Spectra of Lyapunov exponents for the second mutation of the system with rigid pendulum. Parameters Method 1 Method 2 Parameters Method Method ParametersA Method Method λ1 λ2 λ3 λ4 λ1 λ2 λ3 λ4 ω 0.001 0.00621 0.00393 0.00063 0.00952 0.00896 0.00567 0.00091 0.01373 0.0013.64 − − − − 0.001 0.00621 0.00393 −0.00063 −0.00952 0.00896 0.00567 −0.00091 −0.01373 3.64 0.012510.00621 0.00393 0.00393 −0.000630.00651 −0.009520.00996 0.018050.00896 0.00567 0.00567 −0.000910.00940 −0.00.014381373 3.644.94 0.001 − − − − 0.001 0.01251 0.00393 −0.00651 −0.00996 0.01805 0.00567 −0.00940 −0.01438 4.94 0.005160.01251 0.00393 0.00393 −0.006510.00022 −0.009960.00888 0.018050.00745 0.00567 0.00567 −0.009400.00031 −0.00.012811438 4.947.24 − − − − 0.001 0.001 0.00516 0.00393 −0.00022 −0.00888 0.00745 0.00567 −0.00031 −0.01281 7.24 0.00516 0.00393 −0.00022 −0.00888 0.00745 0.00567 −0.00031 −0.01281 For7.24 a small amplitude of pendulum length changes, the behavior of the system is the same as for a rigid pendulum system. For a small amplitude of pendulum length changes, the behavior of the system is the same as FiguresFor a small 15– 17amplitude show quasi-periodic of pendulum behavior length changes, because the they behavior show almost of the thesystem same is behaviorthe same asas for a rigid pendulum system. Figuresfor a rigid2–4 .pendulum The values system. of Lyapunov exponents in Tables3 and6 are very close to each other. This means Figures 15–17 show quasi-periodic behavior because they show almost the same behavior as Figures 15–17 show quasi-periodic behaviorbehavior becausebecause theythey showshow almost the same behavior as Figure 2–4. The values of Lyapunov exponents in Table 3 and Table 6 are very close to each other. Figure 2–4. The values of Lyapunov exponents in Table 3 and Table 6 are very close to each other.

Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 13 of 15 Math. Comput. Appl. 2019, 24, x FOR PEER REVIEW 13 of 15 This means that with a small change in the length of the pendulum, the second system with the first This means that with a small change in the length of the pendulum, the second system with the first mutation of the pendulum behaves as rigid. mutation of the pendulum behaves as rigid. Figures 18–20 show the results for a system with a flexible parametric pendulum. The Figures 18–20 show the results for a system with a flexible parametric pendulum. The Math.corresponding Comput. Appl. spectra2019, 24 ,of 90 Lyapunov exponents can be found in Table 7. 13 of 15 corresponding spectra of Lyapunov exponents can be found in Table 7. In the case of very high stiffness and low damping of the parametric pendulum, the behavior of In the case of very high stiffness and low damping of the parametric pendulum, the behavior of the system is usually quasi-periodic. Phase planes and Poincaré maps presented in Figures 18–20 are thatthe system with a smallis usually change quasi-periodic. in the length Phase of the planes pendulum, and Poincaré the second maps system presented with thein Figures first mutation 18–20 are of very similar to those shown in Figures 2–4. This means that the behavior of a system with a parametric thevery pendulum similar to behavesthose shown as rigid. in Figures 2–4. This means that the behavior of a system with a parametric elastic pendulum is almost the same as that of a rigid counterpart. The computed spectra of Lyapunov elasticFigures pendulum 18–20 isshow almost the the results same for as a systemthat of a with rigid a flexiblecounterpart. parametric The computed pendulum. spectra The corresponding of Lyapunov exponents indicate quasi-periodic behavior. spectraexponents of Lyapunov indicate qu exponentsasi-periodic can behavior. be found in Table7.

Figure 18. Phase planes (gray dots) and Poincaré maps (red, green and blue dots) = 0.05, = 2000 FigureFigure 18.18. Phase planes (gray dots) and and Poincaré Poincaré mapsmaps (red, (red, green green and and blue blue dots) dots) C == 0.050.05,, k = 2000 and = 3.64. andand ω= =3.64. 3.64.

Figure 19. Phase planes and Poincaré maps (colors as above) for =0.05= , = 2000 and =4.94= . FigureFigure 19. 19. PhasePhase planes planes and and Poincaré Poincaré mapsmaps (colors (colors as as above) above) for for =0.05C 0.05,, k =2000 2000and andω =4.944.94. .

FigureFigure 20. 20. PhasePhase planes planes and and Poincaré Poincar émapsmaps (colors (colors as as above) above) for for =0.05C = 0.05,, k = 2000 andandω =7.24= 7.24. . Figure 20. Phase planes and Poincaré maps (colors as above) for =0.05, = 2000 and =7.24. In the case of very high stiffness and low damping of the parametric pendulum, the behavior of TableTable 7.7. SpectraSpectra ofof LyapunovLyapunov exponentsexponents forfor thethe systemsystem withwith elasticelastic pendulum.pendulum. the system is usuallyTable 7. quasi-periodic. Spectra of Lyapunov Phase exponents planes and for Poincarthe systemé maps with elastic presented pendulum. in Figures 18–20 are very similarParameters to those shown in Figures2–4. This meansMethod that the behavior of a system with a parametric Parameters Method elastic pendulum is almost the same as that of a rigid counterpart. The computed spectra of Lyapunov exponents indicate quasi-periodic behavior. Visible differences may be due to early stage dynamics, where both methods may have different convergence factors to the most true values. However, we note that despite this, the type of behavior observed has been verified. Math. Comput. Appl. 2019, 24, 90 14 of 15

Table 7. Spectra of Lyapunov exponents for the system with elastic pendulum.

Parameters Method 1 C λ λ λ λ λ λ ω 1 2 3 4 5 6 0.05 0.00393 0.00075 0.01353 0.01600 0.06176 0.08026 3.64 − − − − 0.05 0.01020 0.00393 0.01600 0.02238 0.06867 0.07771 4.94 − − − − 0.05 0.00393 0.00136 0.01149 0.01600 0.06330 0.07856 7.24 − − − − − Method 2

λ1 λ2 λ3 λ4 λ5 λ6 0.05 0.00567 0.00108 0.01952 0.02308 0.08910 0.11579 3.64 − − − − 0.05 0.01471 0.00567 0.02309 0.03229 0.09907 0.11211 4.94 − − − − 0.05 0.00567 0.00197 0.01658 0.02308 0.09132 0.11333 7.24 − − − − −

5. Conclusions Mathematical models of the system with inertial sliding body, rigid pendulum and its parametric mutations were created. The dynamics of the three systems were examined on the basis of phase planes, Poincaré maps and spectra of Lyapunov exponents. It was observed that the first system consisting of a fixed-length pendulum and an inertial slider behaves quasi-periodically. The results for the second system show that the behavior of the system strongly depends on the amplitude of the periodic change in length around its static elongation. For small pendulum length amplitudes, the behavior is quasi-periodic or chaotic, but if the amplitude is high enough, the system tends to be chaotic, as confirmed by Lyapunov exponents. As for the system containing the parametric pendulum, it has been observed that for the same value of pendulum stiffness the behavior of the system depends on the amount of dispersed kinetic energy. In the case of small values of the damping coefficient, the system easily behaves chaotically, while in the case of high values of the coefficient, it behaves periodically. If the amplitude of the pendulum length change around the static elongation for the second system is very small, the system behaves almost the same as a rigid pendulum system. A similar conclusion can be drawn by comparing the behavior of the third system (the second parametric mutation of the pendulum) with the first system. For high elastic pendulum stiffness value and low damping factor the observed behaviors are similar. Selected methods of computing Lyapunov exponents gave rather similar results and proved to be very useful. The behavior of the examined models based on the obtained values of Lyapunov exponents are consistent with the results obtained from the Poincaré map visualization. It has been observed that for the periodic and chaotic movement, the values of Lyapunov exponents appear to stabilize over time. In the case of quasi-periodic behavior of the spectral system of Lyapunov exponents, they fluctuate with decreasing amplitude.

Author Contributions: Conceptualization, P.O.; methodology, W.S.´ and T.L. and P.O.; software, P.O., W.S.;´ validation, W.S.´ and P.O.; formal analysis, W.S.´ and P.O.; investigation, W.S.´ and T.L. and P.O.; resources, P.O.; data curation, W.S.´ and T.L.; writing—W.S.´ and P.O.; writing—review and editing, P.O.; visualization, W.S.´ and T.L. and P.O.; supervision, P.O.; project administration, W.S.´ and T.L. Acknowledgments: The work was carried out in 2018–2019 as part of the “Research Problem Based Learning” project at the Faculty of Mechanical Engineering at the Lodz University of Technology. Conflicts of Interest: The authors declare no conflicts of interest. Math. Comput. Appl. 2019, 24, 90 15 of 15

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