Integrability Analysis of Chaotic and Hyperchaotic Finance Systems

Home , Lorenz system, Multiscroll attractor

Nonlinear Dyn https://doi.org/10.1007/s11071-018-4370-3

ORIGINAL PAPER

Integrability analysis of chaotic and hyperchaotic ﬁnance systems

Wojciech Szumi´nski

Received: 5 October 2017 / Accepted: 14 May 2018 © The Author(s) 2018

Abstract In this paper, we consider two chaotic tain particular solutions. On the other hand, we show finance models recently studied in the literature. The that for certain sets of the parameters (a, b, c, d, k), first one, introduced by Huang and Li, has a form of when ≤ 0, the hyperchaotic system possesses a poly- three first-order nonlinear differential equations nomial first integral, which can be used to reduce the x˙ = z + (y − a)x, y˙ = 1 − by − x2, z˙ =−x − cz. dimension of the system by one. The second system, called a hyperchaotic finance Keywords Chaotic finance model · Hyperchaotic model, is defined by finance model · Non-integrability · Non-Hamiltonian systems · Numerical analysis · Differential Galois x˙ = z + (y − a)x + u, y˙ = 1 − by − x2, theory z˙ =−x − cz, u˙ =−dxy − ku.

( , , , , ) In both models, a b c d k are real positive parame- 1 Introduction ters. In order to present the complexity of these systems Poincaré cross sections, bifurcation diagrams, Chaos is a very interesting and common phenom- Lyapunov exponents spectrum and the Kaplan–Yorke ena in nature. Having been discovered originally by dimension have been calculated. Moreover, we show Lorenz [1] within the context of atmospheric convec- that the Huang–Li system is not integrable in a class tion, it has found practical applications in various fields ( , , ), of functions meromorphic in variables x y z for all of science, in both theoretical and practical point of ( , , ) real values of parameters a b c , while the hyper- view. Therefore, for the last five decades researchers = chaotic system is not integrable in the case when k c have made a great effort for constructing new chaotic := + ( + − )> and 1 d a d c 0. We give analytic models for chaos-needed applications. Among others, proofs of these facts analyzing properties of the differ- let us mention classical ones: the logistic and the Hénon ential Galois groups of variational equations along cer- maps [2,3], Chua’s circuits [4], the Lorenz-like systems The research has been supported by the National Science introduced by Chen and Lü in [5,6], and more recent Centre of Poland under Grants DEC-2013/09/B/ST1/04130 and systems, studied by Qi and Li in [7–9], the 4D chaotic DEC-2016/21/N/ST1/02477. Duffing system [10], the Chameleon model [11], and B many more [12–19]. W. Szuminski ´ ( ) Among complex dynamical systems exceptional Institute of Physics, University of Zielona Góra, Licealna 9, 65-407 Zielona Gora, Poland role play hyperchaotic ones. These systems, reported e-mail: [email protected] first time by Rössler [20], possess very peculiar prop- 123 W. Szuminski ´ erties. For instance, they have at least two positive question from the economical point of view to avoid Lyapunov exponents and thus, the dynamics of hyper- the chaotic behaviors and make the precise economic chaotic attractor expands in more than one direc- prediction possible. In order to get a quick insight into tion. Since hyperchaotic systems possess usually richer the dynamics of a considered system, we can make dynamics than chaotic ones, researchers have seen its a numerical analysis. Nonetheless, numerical methods great potential for application in engineering, cryp- and techniques, such as Poincaré cross sections, bifur- tography and technology to improve the security of cations diagrams, Lyapunov exponents, power spec- communication. Over the last years, hyperchaotic gen- tra, can be made only for fixed values of parameters erators have been a topic of active search. Except describing a system. For other values, results can be Rössler’s system, many hyperchaotic models have been completely different and, for particular their sets, the found, both in theoretical and in practical ways; see, for system can be even integrable. However, it is techni- instance, recent papers [21–28], and references therein. cally and theoretically impossible to make a numerical Early period linear models of economic cycles were analysis for all values of parameters. To find integrable criticized for their incompatibility with complex real- cases, one needs a powerful method that enables to dis- ity. Their dynamics was too regular to assume that they tinguish values of parameters for a considered system are able to describe properly the complex nature of real is suspected to be integrable. The aim of this paper is economic systems. It was only 1975 when May and to present how, in an effective way, we can perform an Beddington [29,30], as the first, suggested the appli- integrability analysis of complex dynamical systems. cation of nonlinear and chaotic dynamical systems in For this purpose, we will use differential Galois theory. economics and financial markets. In contrary to linear This approach allows, in an analytical way, to prove and stochastic models, nonlinear deterministic models the non-integrability of a considered system in a wide explain aperiodic business cycles and can give rise to class of functions and to distinguish values of parame- chaotic behaviors. Nowadays, in the era of comput- ters for which it is suspected to be integrable. We focus ers, researchers have found chaotic behaviors in vari- our attention mainly on two examples of chaotic and ous known models and new ones with very complex hyperchaotic finance systems, recently studied in the dynamic are being proposed. For instance, let us men- literature. tion the forced van der Pol model [31,32], the Kaldo- The rest of this paper is organized as follows. In rian model [33,34], various types of the IS-LM model Sect. 2, descriptions of systems and their dynamics developed by Hick and Hansen [35–37], Goodwin’s analysis for specified values of parameters are given. accelerator model [38,39], and recently, the Keyne- For this purpose, we calculate Poincaré cross sections, sian model [40], the Cournot–Puu system [41], the bifurcation diagrams, Lyapunov exponents and fractal Huang–Li model [42,43] and its generalizations to 4D dimension. The integrability analysis of investigated hyperchaotic systems [22,24,28], to cite just a few. We models by means of the differential Galois approach also highly recommend this year special issue, enti- is made in Sect. 3. In Sect. 4, final conclusions are tled Dynamics Models in Economics and Finance, pub- drawn. In order to make this article self-contained, we lished in [44]. have added “Appendix” section with the basic defi- The presence of parameters is typical for many mod- nitions and facts concerning integrability, differential els of economic processes. For example, in economic Galois theory and its application to integrability inves- growth models, they may represent tools for influenc- tigations. ing the economy, while the aim of the analysis is to find such quantities that would lead to the optimal path of growth. However, if the analyzed model has chaotic dynamics, the matter is essentially complicated. The 2 Description of the systems and their dynamical high sensitivity of chaotic system to a change the ini- behaviors tial conditions makes it impossible to predict the effects of economic decisions in a long time scale. Therefore, it 2.1 The Huang–Li financial model is important to ask whether there exists any set of values of parameters for which dynamics is regular and Recently, the Huang–Li system, known also as a a considered system is integrable. It is the significant chaotic finance model, has been investigated inten- 123 Integrability analysis of chaotic

Quasi-periodic motion

Chaotic Periodic motion motion

Fig. 1 Poincaré cross section of system (1) with a = b = c = 0, on the surface y = 0 sively. It is given by the following equations ⎧ ⎨⎪x˙ = z + (y − a)x, y˙ = 1 − by − x2, (1) ⎩⎪ z˙ =−x − cz, where (x, y, z) are time-dependent variables and (a, b, c) are real nonnegative parameters. Here, x represents the interest rate, y is the investment demand, and z is the price index. Parameters (a, b, c) denote the saving amount, the cost per investment and the elasticity demand of commercial markets, respectively. The complex behavior of system (1) was ﬁrst noted by Ma and Chen in 2001 [45,46]. Then many papers were published where the dynamics of this model has been investigated by means of various methods and techniques, such as Lyapunov exponents and bifurcation diagrams [22,47]; synchronizations with linear and nonlinear feedbacks [48,49], adaptive [50], sliding mode [51] and passive [51] control methods [51]; control via linear, speed and time-delay feedbacks [52–54].

2.1.1 Poincaré sections and phase portraits Fig. 2 Periodic, quasi-periodic and chaotic particular phase = = = To get an idea about complexity of the system dynam- curves of system (1) for parameters a b c 0. Points of intersections with Poincaré cross-sectional plane ics, we made two Poincaré cross sections, which are are depicted by small black bullets. a Periodic trajectory shown in Figs. 1 and 3. Points creating presented pat- with (x0, y0, z0) = (0.49, 0, 0). b Quasi-periodic trajectory ( , , ) = ( . , , ) terns were obtained as traces of intersections of tra- with x0 y0 z0 0 8 0 0 . c Chaotic trajectory with ( , , ) = (− . , , − ) jectories calculated numerically with suitably chosen x0 y0 z0 3 5 0 1 123 W. Szuminski ´

Fig. 3 Poincaré cross section of system (1) for parameters a = 0.1, b = 0.01, c = 0.67, on the surface y = 0 plane of section y = 0. The coordinates on this plane are (x, z). The first Poincaré section, visible in Fig. 1, was made for zeroth values of the parameters. As we can see, even in this simple case, the system reveals very complex dynamics. In fact, for a = b = c = 0sys- tem (1) converts into the famous Nosé–Hoover oscillator [55,56]. Poincaré section visible in Fig. 1 is similar to that for conservative systems, i.e., we can observe the coexistence of periodic, quasi-periodic and chaotic motions. For easier understanding particular orbits corresponding to these different types of motion have been marked at the Poincaré section plane, while their cor- Fig. 4 Successive magnifications of the small selected region in responding phase curves, with depicted points of inter- Fig. 3 showing fractals structure of system (1) sections with Poincaré cross-sectional plane, are visible in Fig. 2. In particular, we do not observe appearance fractal structure for dissipative system. Indeed, Fig. 4 of an attractor, which is common in the case of sys- shows successive magnifications of the small selected tems with dissipation. This is due to the fact that for region in Fig. 3, which displays the hidden layers pre- zeroth values of the parameters, the divergence of (1) serving self-similarity. is just y and in the considered example its mean value There is more detailed explanation about the differ- is almost zero to a high precision of numerical calcu- ences between the Poincaré cross sections of conser- lations. Thus, it is natural to ask whether system (1) vative and dissipative systems the interested reader can has an invariant measure. However, direct calculations find, for example, in the book [57]. have not yield any results, so let us leave this question as the open problem. Poincaré cross section completely changes when we 2.2 A hyperchaotic finance model chose nonzero values of the parameters. Figure 3 corresponds to dynamics of system (1) with a = 0.1, b = One of generalizations of the Huang–Li system is called 0.01, c = 0.67. Since the divergence is y −a −b−c = a hyperchaotic finance model. This system introduced 0, volume of the phase space is contracted, and we got in [22] is described by a set of four nonlinear differential the expected shapely elegant strange attractor of the equations 123 Integrability analysis of chaotic ⎧ ⎪ ⎪x˙ = z + (y − a)x + u, He found that for many systems, which exhibit bifurca- ⎨⎪ y˙ = 1 − by − x2, tions, the intervals between successive thresholds can ⎪ (2) be determined from the formula ⎪z˙ =−x − cz, ⎩⎪ 1 u˙ =−dxy − ku, (k + − k ) ≈ (k − k − ), n 1 n δ n n 1 (3) ( , , , , ) where a b c d k are real nonnegative parameters. where kn is the value for which n bifurcations appears, , , Here, x y z have the same meaning as in the Huang– and kn+1 is the value for which the next bifurcation Li model, while u is an additional state variable that takes place. The number represents the average profit margin. Since the system k − k − is four dimensional, it is rather difficult to interpret its δ = lim n n 1 = 4.6692016, (4) n→∞ k + − k Poincaré cross sections. Therefore, in order to present n 1 n complexity of this model one can study its bifurca- is the universal constant called Feigenbaum’s number. tion diagrams, Lyapunov exponents’ spectrum and the Relation (3) implies that difference between successive Kaplan–Yorke dimension. bifurcations (kn+1 − kn) tends asymptotically to zero, and hence, value of the control parameter k approaches a finite limit. For our system, this critical value is found 2.2.1 Bifurcation diagram to be =∼ . . Bifurcation diagrams provide an important view to a kc 1 6711 system dynamics by plotting the state variable as a This means that for k > kc the motion never repeats function of a bifurcation parameter [58]. More pre- itself and is chaotic. cisely, a bifurcation diagram of a dynamical system x˙ = v(x, k), where k is a control parameter, illustrates 2.2.2 Lyapunov exponents’ spectrum and the how the limit set (fixed points, periodic trajectories, Kaplan–Yorke dimension chaotic attractors) changes with a change of k.Fig- ure 5a shows such a global bifurcation diagram. This Although the period-doubling route to chaos is seen diagram plots the extremal values of the system vari- at bifurcation diagram in Fig. 5a, the hyperchaotic able x(t) = x , satisfying x˙(t) = 0, where initial extr nature of this system can be recognized only by their transient motions have been omitted. For fixed param- Lyapunov exponents. According to the chaos theory, eters b = 0.1, c = 1.45, d = 0.18, a = 2.36 and initial Lyapunov exponents measure the exponential rates condition (x , y , z , u ) = (1, 2, 0.5, 0.5),weshow 0 0 0 0 of propagation of nearby trajectories (orbits) in the how the change of value of parameter k in the interval phase space. For four-dimensional autonomous sys- [0, 2] affects the dynamics of the system. As we see, the tems, chaos appears when at least one Lyapunov expo- periodic motion diverges and splits as k decreases and nent is positive, whereas hyperchaos is manifested by finally becomes chaotic. Indeed, Fig. 5b shows magnifi- at least two positive Lyapunov exponents. cation of the lower right area of the bifurcation diagram, Figure 6 presents a spectrum of the Lyapunov expo- in which we can observe first successive bifurcations nents’ of system (2) versus parameter k calculated by that lead the system to chaotic behavior through the Wolf’s algorithm [59], over interval k ∈[0, 2].At period-doubling route. first sight, we observe that the Lyapunov exponents’ In the case of continuous dynamical systems, the spectrum corresponds well to the bifurcation diagram. period-doubling route starts with limit cycle behavior Let λ denote the Lyapunov exponents sorted in the of the system. As a control parameter is changed, a i decreasing order. Then, the dynamics of (2) can be sum- periodic solution may become unstable and gives birth marized in the following way. to a period two cycle. In Fig. 5b, we pointed the values of parameter k (to 4 significant figures) at which the 1. For periodic motion: λ1,2,3 = 0,λ4 < 0. subsequent bifurcations appeared. 2. For quasi-periodic motion: λ1,2 = 0,λ3,4 < 0. The period-doubling mechanism, that is one of the 3. For chaotic motion: λ1 > 0,λ2 = 0,λ3,4 < 0. scenarios of transition to chaos, was intensively stud- 4. For hyperchaotic motion: λ1,2 > 0,λ3 = 0, ied by physicist and mathematician M. J. Feigenbaum. λ4 < 0. 123 W. Szuminski ´

(a) Fig. 6 Lyapunov exponents’ spectrum of system (2) versus parameter k for b = 0.1, c = 1.45, d = 0.18, a = 2.36. For a better readability, the green-color exponent has been shifted up by 0.7. (Color ﬁgure online)

While Lyapunov exponents measure sensitivity of a dynamical system to initial conditions, the Kaplan– Yorke dimension (or the Lyapunov dimension) of an attractor measures its complexity [60]. The dimension of an attractor is defined in terms of Lyapunov expo- λ = + L λ /|λ | nents i ,byformulaDKY L j=1 i L+1 , where L is a maximum integer value of i, such that L λ ≥ L+1 λ ≤ i=1 i 0, and i=1 i 0. Figure 7 shows a global spectrum of the Kaplan– Yorke dimension of system (2), versus parameter k ∈ [0, 2], and its magnification over the interval, where (b) the hyperchaotic behavior takes place. Using quantities (5), or directly from Fig. 7b, we can conclude Fig. 5 Bifurcation diagram of system (2) versus k for parameters that a maximum value of the Kaplan–Yorke dimen- a = . , b = . , c = . , d = . a 2 36 0 1 1 45 0 18. Global view for =∼ . k ∈[0, 2]. b Magnification presenting the period-doubling route sion of the system is DKY 3 067. For compari- to chaos son, the famous hyperchaotic Rössler system has an ∼ attractor with DKY = 3.005, whereas a hyperchaotic attractor found in [61] possesses a fractal dimension While the hyperchaos was not noticeable at the bifur- D =∼ 3.198. cation diagram, it is clearly visible at the Lyapunov KY As we can notice, the hyperchaotic behavior of sys- exponents’ spectrum. Indeed, if k belongs to inter- tem (2) is not so noticeable as the one studied in [61], val [0.158, 0.209], then two values of the Lyapunov where the maximal Lyapunov exponent has been found exponents are positive and the system manifests hyper- to be λ =∼ 29.79 and D =∼ 3.198. However, a chaotic behaviors. They reach the largest values at max KY model studied in the mentioned article was especially k =∼ 2.03, where created to exhibit very strong hyperchaotic behavior ∼ ∼ ∼ ∼ λ1 = 0.0481,λ2 = 0.0246,λ3 = 0,λ4 = −1.0804. and it has no economic interpretation. (5)

123 Integrability analysis of chaotic

3.07 3.0

3.06

2.5 3.05

3.04 2.0 3.03

3.02 1.5

3.01

1.0 3.00 0.0 0.5 1.0 1.5 0.16 0.17 0.18 0.19 0.20 0.21 k k (a) (b)

Fig. 7 Spectrum of the Kaplan–Yorke dimension of system (2) versus parameter k for a = 2.36, b = 0.1, c = 1.45, d = 0.18. a Global view with k ∈[0, 2]. b Magniﬁcation of the hyperchaotic regime

T n 3 Integrability analysis of the models x˙ = v(x), x = (x1,...,xn) ∈ R , (6) − The complex behaviors of models (1)–(2) apparent via is B-integrable if there exist n k vector fields (sym- ( ) = v( ), ( ),..., ( ) the Poincaré cross sections, the bifurcation diagram metries) u1 x x u2 x un−k x , which [ , ]= and Lyapunov’s exponents spectrum, suggest their non- pairwise commute ui u j 0, and k functions F ,...,F integrability. But these numerical evidences of non- 1 k, which are common first integrals of all vec- ,..., integrability were obtained for chosen values of param- tor fields u1 un−k. eters. For other values, plots can be completely differ- It can be shown that if system (6)isB-integrable, ent and, for particular their sets, these systems can be then it is integrable by quadrature. That is, all its solu- even integrable. However, it is technically and theo- tions can be obtained by means of finite sequence of retically impossible to make a numerical analysis for algebraic operations, superposition and inversion of all values of parameters. To find integrable cases, one functions and by calculation of integrals. Basic facts needs a powerful tool that enables us to select these concerning the integrability and in particular a more values of parameters for which the considered systems formal definition of the B-integrability can be found in are suspected to be integrable. “Appendix” section. For Hamiltonian systems for which we have a The aim of this letter is to check whether there exist precise notion of integrability in the sense of Liou- any sets of values of the parameters for which sys- ville, there are many approaches to the integrabil- tems (1) and (2)areB-integrable. The main results are ity studies: Hamilton–Jacobi theory and separation of formulated in the following two theorems. variables, perturbation theory, splitting of separatrices and Melnikov’s method, Birkhoff’s normalization, and Theorem 1 The Huang–Li nonlinear financial model recently, Morales–Ramis theory based on the differen- defined by (1) is not B-integrable in a class of meromor- ( , , ) tial Galois approach. In contrast, for non-Hamiltonian phic functions of variables x y z for all real values ( , , ) systems even there is no commonly accepted defi- of parameters a b c . nition of integrability. It seems that quite useful is the so-called B-integrability. We say that a given n- Theorem 2 The hyperchaotic finance system defined dimensional system by (2) is not B-integrable in a class of meromorphic functions of variables (x, y, z, u) for values 123 W. Szuminski ´ of parameters (a, b, c, d), satisfying k = c and be found in “Appendix” section and in cited references 1 + d(a + d − c)>0. therein. In particular, for first applications of differential Galois theory to non-Hamiltonian systems, please To prove these theorems, we investigate variational consult papers [67,68]. equations of the considered systems along certain non- stationary solutions and we study their differential Galois groups. This approach of finding necessary con- 3.1 Proof of Theorem 1 ditions for integrability in a framework of differential Galois theory was mostly used for Hamiltonian sys- System (1) possesses the invariant manifold tems, and it is frequently called Morales–Ramis theory. M = (x, y, z) ∈ C3 x = z = 0 . (8) Theorem 3 (Morales–Ramis [62]) Assume that a Hamiltonian system is meromorphically integrable in Indeed, Eq. (1) restricted to N reads as follows: the Liouville sense in a neighborhood of the analytic x˙ = 0, y˙ = 1 − by, z˙ = 0. (9) phase curve . Then, the identity component of the differential Galois group of the variational equations Solving these equations, we obtain our particular along is Abelian. solution ϕ(t) = (0, y(t), 0).LetX =[X, Y, Z]T be the variations of x =[x, y, z]T ; then, the first-order Thanks to this approach, new integrable cases have variational equations along ϕ(t) take the form been found; see for instance [63–65]. d ∂v(x) If we restrict ourself to the B-integrability, then we X = M(t) · X, M(t):= (ϕ(t)), (10) dt ∂x have an elegant generalization of Morales–Ramis the- ( ) ory for non-Hamiltonian systems. Namely, with sys- where the explicit form of matrix M t is given by ⎡ ⎤ tem (6) we can associate its cotangent lift, i.e., a Hamil- y − a 01 tonian system defined by the following Hamiltonian M(t) = ⎣ 0 − b 0 ⎦ , function − 10− c n and y = y(t) satisfies (9). Since the particular solution H = pi vi (x), (7) i=1 corresponds to a motion along y-axis, the equations for X and Z form a subsystem of normal variational where x = (x ,...,x ) and p = (p ,...,p ) are 1 n 1 n equations canonical coordinates on a symplectic manifold M = C2n. In paper [66] Ayoul and Zung showed that if orig- X˙ y − a 1 X = , (11) inal system (6)isB-integrable, then a lifted system Z˙ − 1 − c Z generated by Hamiltonian function (7) is integrable in that can be expressed as one second-order differential the Liouville sense. Hence, for both Hamiltonian and equation non-Hamiltonian systems, we have the same necessary integrability condition, i.e., the identity component of X¨ + (a − c − y)X˙ + (ac + (b − c)y) X = 0. (12) the differential Galois group of variational equations Next, by means of the change of the independent must be Abelian. We summarize the above facts by variable the following theorem that gives necessary integrabil- −1 + by(t) ity conditions for non-Hamiltonian systems. t −→ s = , with b = 0 (13) b2 Theorem 4 (Ayoul–Zung[66]) Assume that system (6) and using the chain rule for transformation of deriva- is meromorphically B-integrable, then the identity com- tives ponent of the differential Galois group of variational d d d2 d2 d equations along a particular non-equilibrium solution =˙s , =˙s2 +¨s , dt ds dt2 ds2 ds ϕ(t) is Abelian. we can rewrite Eq. (12)as A more detailed description of differential Galois the- d X + P(s)X + Q(s)X = 0, ≡ . (14) ory and its application to the integrability studies can ds 123 Integrability analysis of chaotic

Explicit forms of the coefficients P(s) and Q(s) are Theorem 5 (Morales–Ruiz [62]) The identity compo- the following nent of the Galois group of equation (18) is Abelian if and only if numbers (p, q) defined by 1 + b(b − a − c) P(s) = + , 1 2 b s := κ + μ − 1, := κ − μ − 1 b − c + abc b − c p q (20) Q(s) = + . (15) 2 2 b3s2 bs ∗ ∗ belong to (N ×−N ) ∪ (−N × N). The classical change of the dependent variable ⎡ ⎤ s Let us assume that the considered system is integrable. 1 Then, according to Theorem 5 the product p · q should X = w exp ⎣− P(l)dl⎦ , (16) 2 always be negative. In our case, however, for (κ, μ) s0 given in (19), we obtain the following equality transforms (14) into its reduced form 1 1 p · q = κ2 − κ − μ2 + = . (21) 2 2 4 2 2 1 d + b d − b − 4b 4 b w (s) − − + w(s) = 0, 4 2b2s 4b4s2 Thus, p · q > 0 for all real b > 0, and this ends the (17) proof. where d := ab − bc − 1. Here, we should underline one significant fact. Namely, respective transformations (13) and (16) 3.2 Proof of Theorem 2 change in general a whole differential Galois group of variational equations (10). However, the clue is that In order to simplify further calculations, let us write they do not affect the identity component of the group; z = v − u, where v is a new variable. System (2)after see, e.g., [62]. Thus, following Theorem 4, in order this linear change of the variable is given by ⎧ to prove the non-integrability of our original nonlin- ⎪ ⎪x˙ = v + (y − a)x, ear system (1), it is sufficient to show that the identity ⎨⎪ component of the differential Galois group G of varia- y˙ = 1 − by − x2, ⎪ (22) tional equations (10), and thus, its rationalized-reduced ⎪v˙ = c(u − v) − ku − x(1 + dy), ⎩⎪ form (17) is not Abelian. In order to check whether it is u˙ =−dxy − ku. Abelian or not, we usually apply the so-called Kovacic algorithm [69]. This algorithm classifies possible solu- We restrict further analysis to case k = c, in order to tion forms of second-order differential equations with find a particular solution in an explicit form. Indeed, in rational coefficients and their corresponding differen- this case system (22) has the two-dimensional invariant tial Galois groups. manifold However, in the considered case, the situation is M = (x, y,v,u) ∈ C4 x = ,v= , much simpler. In Eq. (17), we immediately recognize 0 0 (23) the Whittaker equation and its restriction to N is given by 2 1 κ 4μ − 1 w (s) − − + w(s) = 0, (18) x˙ = 0, y˙ = 1 − by, v˙ = 0, u˙ =−cu. (24) 4 s 4s2 Solution of (24) determines our particular solution with parameters ϕ( )) = ( , ( ), , ( )) √ t 0 y t 0 u t . The variational equations + 2 2 − 2 along ϕ(t) take the following form κ := d b ,μ:= d 4b . 2 2 (19) ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 2b 2b X˙ y − a 010 X This equation has one regular singularity at z = 0 ⎢Y˙ ⎥ ⎢ 0 − b 00⎥ ⎢Y ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ , (25) and one irregular singularity at z =∞. The differential ⎣V˙ ⎦ ⎣− 1 − dy 0 − c 0 ⎦ ⎣V ⎦ Galois group of the Whittaker equation is described by U˙ − dy 00− c U the following. 123 W. Szuminski ´ where [X, Y, V, U]T are the variations of [x, y,v,u]T of (x, y, z, u) and equaling to zero its all coefficients, and y = y(t) satisfies (24). The normal part of this we obtain the system of equations for unknown coeffi- system containing variables (X, V ) can be written as cients of the requested first integral F. Its solutions give follows restrictions for the values of parameters and determine X¨ + (a + c − y) X˙ + (ac + (b − c + d)y)X = 0. the forms of first integrals which are given below. In the case when (26) 1 a =− , d = k, b, k, c = arbitrary, (30) Next, we perform change of independent vari- c able (13) that transforms (26) to rational equation (14), system (2) possesses a linear first integral of the form with coefficients 1 + b(b − a − c) F = k(cx + z) + cu. (31) P(s) = + , 1 2 b s For the values k = c and b − c + abc + d b − c + d √ √ Q(s) = + . (27) 17 2 14 14 √ b3s2 bs a =− √ , c = , b = , d = 14, Then, by means of change of dependent vari- 3 14 3 6 able (16), we write it in the form of the Whittaker equa- (32) κ μ tion (18) with parameters and defined by it has the cubic first integral √ √ b(a + c − c + 2d) − 1 F = 3 + 2( + ) + [ − ] κ = , 98x 7 14x u 29z 14x 18√ 6 14y 2b2 + 7y2 + 28z(u + z) + (u + z)[7y( 14y − 12) 1 + b(a2b − 2a(1 + bc) + b(c2 − 4) + 2(c − 2d) √ μ = . + ( + ( + ))]. 2b2 2 14 9 7z u z (33) (28) Whereas, for k = c and √ √ Application of Theorem 5 gives the following equal- 7 3 10 10 √ ity a =− √ , c = , b = , d = 10, 2 10 4 4 1 p · q = κ2 − κ − μ2 + = , (29) (34) 4 b2 := + ( + − ), it possesses the quartic first integral where 1 d a d c has been introduced. √ √ From this, we conclude that as long as >0the F = 50x4 + 10 10x3(u + 11z) + 5x2[16 − 8 10y identity component of the differential Galois group of + 10y2 + (u + z)(u + 61z)] variational equations (26) is not Abelian, and thus, the √ + 2x(u + z)[5y( 10y − 8) necessary integrability condition is not satisfied. This √ + ( + ( + ))] ends the proof. 10 8 √15z u z +[5y2 − 4 10y + 2(4 + 5z(u + z))](u + z)2.

3.3 First integrals of the hyperchaotic finance model (35) It is easy to verify that for all sets of the parame- In the previous subsection, we have shown that the ters (30), (32) and (34) with k = c the value of is hyperchaotic finance model is not B-integrable for equal to zero. Thus, the necessary integrability condi- k = c and = 1 + d(a + d − c)>0. On the other tion is satisfied, as it should be. hand, if such inequality is not satisfied, then there is no integrability obstacles and system (2) can possesses first integrals at least for certain values of parameters 3.4 Reduction of the hyperchaotic finance model (a, b, c, d, k). So let us look for such first integrals. Since the right-hand sides of (2) are polynomial in In the previous subsection, we have shown that for cer- variables (x, y, z, u), we search for a polynomial first tain sets of parameters (a, b, c, d, k), the hyperchaotic integral F(x, y, z, u) ∈ R[(x, y, z, u)] with undeter- system possesses polynomial first integral. Thus, for mined coefficients. Postulating the degree of F, com- prescribed values of the parameters we can use it to puting the Lie derivative Lv(F) that is a polynomial reduce the dimension of the system by one. So, let us 123 Integrability analysis of chaotic

Fig. 8 Bifurcation diagram of reduced model (36) versus k for b = 0.01, c = 0.71, f = 0 put the values of parameters a =−1/c, d = k,for which system (2) has linear first integral (31). Then, determining variable u from integral F = f , where f denotes a constant level of it, we reduce model (2)to the following one ⎧ ⎪ 1 f − k(cx + z) ⎨⎪x˙ = z + y + x + , c c 2 (36) ⎪y˙ = 1 − by − x , ⎩⎪ z˙ =−x − cz. In order to get a quick insight into the dynamics of this three-dimensional system, we make the bifurcation diagram and the Poincaré cross section, which are showninFigs.8 and 9. As we can notice, for parameters b = 0.01, c = 0.71, k = 0.02, f = 0, system (36) still reaches very complex dynamics. Like in the Huang– Fig. 9 Poincaré cross section of reduced model (36)forb = Li system, we obtain shapely elegant strange attractor 0.01, c = 0.71, k = 0.02, f = 0, on the surface y = 0. a Global of the fractal structure; see especially Fig. 9b showing portrait. b Magnification of the selected region hidden layers. Indeed, if we put a =−1/c in (1), and f = k = 0in(36), then the reduced hyperchaotic integrable. Thus, if such inequality is not satisfied, model (36) coincides with chaotic one (1). Similar then (36) may possesses an additional first integral. reductions can be made for other sets of the param- However, direct search for a polynomial one up to the eters for which there exist cubic (33) and quartic (35) fourth degree in variables x did not yield any results. first integrals, respectively. However, its final form will be more complicated because of nonlinear dependence of all variables in first integrals. 4 Conclusions It is worth mentioning that system (36)onlevel f = 0 carries particular solution (9). Making the ana- It seems that analysis of a differential Galois group of logues calculations as in Sects. 3.1–3.2, we conclude variational equations gives one of the strongest known that as long as 1 − k/c > 0, system (36) is not B- obstructions to integrability. This method only requires 123 W. Szuminski ´ knowledge of a particular solution that is not an equi- 5 Appendix: Integrability and basics facts from librium position. Indeed, in order to find such a solution general theory for the hyperchaotic finance model we put k = c.Even though it has some limitation, it is weaker than many of In this section, we give basic facts concerning the assumptions required in other methods and the result notion of integrability, differential Galois theory and of application of this method is the non-integrability its application to investigations of integrability. A more proof of a considered system in a wide class of func- detailed introduction to this theory can be found, e.g., tions meromorphic in variables. Furthermore, if a sys- in [62,70,71]. tem depends on certain parameters (e.g., as the models investigated in this paper), then we can usually prove its non-integrability for almost all values of parame- 5.1 Integrability ters except some finite set of values. In this way, new integrable cases can be found. Let us consider a system of differential equations

We showed the application of this approach as exem- d m x = v(x), x = (x ,...,xn) ∈ U ⊂ R , (37) plified by intensively studied nonlinear models. We dt 1 proved that the Huang–Li finance model is not inte- where v(x) = (v1(x),...,vn(x)) is a smooth vec- grable in a class of functions meromorphic in variables tor field in the considered domain U. Evolution of a ( , , ) ( , , ) x y z for all real values of parameters a b c .Inthe dynamical system is given by solutions of these equa- case of its generalization, called hyperchaotic finance tions with certain initial conditions x(t ) = x that can = 0 0 model (2), we proved its non-integrability for k c be chosen arbitrarily. However, general solutions are = + ( + − )> ≤ and 1 d a d c 0. Moreover, for 0, explicitly known only for few systems, such as simple we found linear, cubic and quartic polynomial first inte- mathematical pendulum, damped harmonic oscillator ( , , , , ) grals for certain sets of parameters a b c d k . or standard Kepler problem. As it was observed a long Although some values of the parameters have no time ago, a significant role in the study of dynamical < direct economic interpretation (a 0), it is very sur- systems is played by first integrals or other invariant prising that the four-dimensional hyperchaotic finan- quantities remaining constant on all solutions, which cial model possesses certain first integrals, which allow help to find their explicit solutions. Let us recall that a to reduce it to a three-dimensional chaotic system. non-constant function F(x) : Rn → R is called a first Since in the direct search method we restricted our- integral of (37)ifF(x(t)) = const for all solutions selves to the polynomial first integrals, it is natural x(t). In other words, F(x) is a conservation law for to ask about the existence of Darboux’s polynomials, system (37). The invariance of F(x) with respect to the which can be very useful to construct rational first inte- evolution described by (37) (or, mathematically, with grals, and thus to further reduction of the hyperchaotic respect to the flow of system (37)) can be expressed for finance model. Let us leave this question as the open differentiable function in the form problem. n ∂F Lv(F):= vi = 0, (38) Acknowledgements I would like to give special thanks to ∂xi Andrzej J. Maciejewski and Maria Przybylska for their active i interest in publication of this paper. where Lv is the Lie derivative along vector field v(x). The existence of a first integral implies the relation Compliance with ethical standards between coordinates Conflict of interest The author declares that the research was F(x1,...,xn) = F(x1(0),...,xn(0)) = const, conducted in the absence of any conflict of interest. which, in principle, allows to eliminate one variable, Open Access This article is distributed under the terms of e.g., xn = ψ(x1,...,xn−1) to reduce a dimension of the Creative Commons Attribution 4.0 International License F ,...,F (http://creativecommons.org/licenses/by/4.0/), which permits a considered system by one. If we have 1 k unrestricted use, distribution, and reproduction in any medium, first integrals, which are functionally independent, then provided you give appropriate credit to the original author(s) and solutions of system (37) lie on n−k-dimensional hyper- the source, provide a link to the Creative Commons license, and indicate if changes were made. 123 Integrability analysis of chaotic surface in Rn given by a common level of constant val- Definition 1 (Liouville) A Hamiltonian system with m ues degrees of freedom is completely integrable iff it has ,..., n m first integrals F1 Fm, whose Poisson’s brackets M f ,..., f := x ∈ R | F1(x) = f1,...,Fk(x)= fk . 1 k vanish {Fi , Fj }=0, for every i, j = 1,...,m, and Assuming, without loss of generality, that we can use are functionally independent. ˜ = ( ,..., ) x x1 xn−k−1 as coordinates on the manifold In the case of non-Hamiltonian systems, there is no M ,..., , we reduce system (37) to the following one f1 fk commonly accepted definition of integrability. Below ˙ we give two of them. x˜ = v˜(x˜ , f ), f = ( f1,..., fk), (39) v˜i (x1,...,xn−k−1, f1,..., fk) = vi (x1,...xn), Definition 2 (Jacobi) We say that system (37)isinte- grable in the Jacobi sense iff it admits n−2 functionally where i = 1,...,n−k−1, and thus x = x (x, f ),for j j independent first integrals and differential m-form j = n − k,...,n. In particular, if for n-dimensional vector field we know n − 1 first integrals, which are ω = h(x)dx1 ∧···∧dxn, (44) functionally independent, then we can reduce sys- such that tem (37) to only one first-order equation n ∂( v ) (ω) = h i ∧···∧ = . d =˜v ( ), Lv dx1 dxn 0 (45) x1 1 x1 (40) ∂xi dt i=1 that can be easily solved by one quadrature Equation (45) expresses the fact that ω is an invariant quantity with respect to system (37). In the old litera- dx1 = t, (41) ture, the invariant m-form is called the Jacobi last mul- v˜ (x , f ) 1 1 tiplier. This integrability definition is frequently used in and it gives a solution in an implicit form t = γ(x1). non-holonomic mechanics; see, e.g., [68]. Other defi- If we assume that a function γ is invertible, then we nition of integrability was proposed by Bogoyavlensky −1 can determine x1 = γ (t), and other variables can be [72,73]. determined from the algebraic operations such as addi- tion, subtractions or calculations of roots. In this situ- Definition 3 (Bogoyavlensky) We say that a given n- ation, we say that the considered system is integrable dimensional system (37)isB-integrable if there exist by quadratures. 1 ≤ k ≤ n functionally independent first integrals F ,...,F − Suppose that system (37) is a Hamiltonian one. 1 k, and n k symmetries, i.e., vector fields ( ) = v( ), ( ), . . . , ( ) Then, n = 2m and v = vH is a Hamiltonian vec- u1 x x u2 x un−k x such that tor field generated by a smooth scalar function H : [ui , u j ]=0, and Lu (Fi ) = 0, (46) R2m → R, called Hamiltonian. In this context, we con- j sider R2m as a linear symplectic manifold parametrized for 1 ≤ i ≤ k, 1 ≤ j ≤ n − k. by means of canonical coordinates x = (q, p), where In this definition, the second condition means that q = (q ,...,q ) and p = (p ,...,p ). It is equipped 1 n 1 n functions F ,...,F are common first integrals of all with the standard symplectic form 1 k vector fields u1,...,un−k. It can be shown that B- m integrability for non-Hamiltonian systems is similar to ω = ∧ = ∧ , dq d p dqi d pi (42) the integrability in the Liouville sense for Hamiltonian i=1 mechanics; see, e.g., [71]. and the equations of motion vH generated by H are given by 5.2 Differential Galois approach to integrability m ∂H ∂ ∂H ∂ v = − . H ∂ ∂ ∂ ∂ (43) In order to connect the integrability with properties of = pi qi qi pi i 1 differential Galois theory, we have to assume that sys- For Hamiltonian systems, we have the well-known tem (37) is considered over the field of complex func- notion of integrability in the sense of Liouville. tions. More precisely, we assume that the phase space is 123 W. Szuminski ´ a complex manifold M = Cn and the time is a complex not necessarily elements of F. If we add to the field F, variable, i.e., n linearly independent solutions of (48), then the new d differential field L is called the Picard–Vessiot exten- x = v(x), x ∈ Cn, t ∈ C. (47) F G = (L/F) dt sion of . The differential Galois group G of linear system (48) is the group of all automorphisms Most of differential equations describing evolution of the Picard–Vessiot extension that leave elements of of physical systems are nonlinear, and thus, it is very F fixed and commute with differentiation. It can be difficult to analyze them. However, suppose that we shown that G is a linear algebraic subgroup of the group know a particular solution t → ϕ(t), then we can make GL(n, C). As a set group G consists of a certain num- a lineralization of system (47) around this solution. In ber of separated parts, one of them contains the iden- other words, we calculate variational equations along tity matrix and it is called the identity component G0. ϕ(t), which read as follows: If f (g(X)) = f (X) for every element g of the differ- d ∂v X = M(t) · X, where M(t) = (ϕ(t)). (48) ential Galois group of linear system (48), then we say dt ∂x that f is an invariant of G. The reason why properties By means of the variational equations, we investi- of the differential Galois group of variational equations gate the behavior of solutions of nonlinear system (37) are connected with the integrability is as follows. around the phase curve corresponding to a particular solution ϕ(t). Since the times of Henri Poincaré, it has Lemma 2 If system (47) has k functionally indepen- been well known that if an original system has an ana- dent meromorphic first integrals in a neighborhood of lytic first integral, then variational equations possess a phase curve , then the differential Galois group G of first integral that is a polynomial in variables X. variational equations (48) along has k functionally independent rational invariants. Lemma 1 If F(x) isafirstintegralofsystem(37), then the first non-vanishing term of its Taylorexpansion The relation between the presence of first integrals of F(ϕ(t) + X) = fm(X) +···, where fm(X) = 0 is a a nonlinear system and invariants of a certain group homogeneous polynomial in X of degree m ≥ 0,isa related to variational equations was first noticed by first integral of variational equations (48). Ziglin for monodromy group, see [74]. Later, Morales and Ramis extended this observation to the differential Furthermore, it is not difficult to prove that if F(x) Galois group that contains the monodromy group. For is a meromorphic first integral of (47), then the vari- Hamiltonian systems and integrability in the Liouville ational equations have a rational and homogeneous in sense, this relation is particularly elegant. For these variables X first integral. Moreover, Ziglin’s lemma systems, G is an algebraic subgroup of the symplectic (see [74]) guaranties that if system (47) has k function- group Sp(2n, C) and the abelianity of the Lie algebra ally independent and commuting first integrals, then of first integrals implies abelianity of the Lie algebra also variational equations (48) have the same number of the differential Galois group. This is more precisely of functionally independent and commuting first inte- stated in the following theorem. grals. These observations show that integrability of lin- earized system is related to integrability of an original, Theorem 6 (Morales–Ramis) Assume that a Hamilto- nonlinear model. It means that in order to formulate nian system is meromorphically integrable in the Liou- necessary integrability conditions for system (47), we ville sense in a neighborhood of an analytic phase can use its lineralization (48). With linear system (48), curve . Then, the identity component of the differ- we can connect a group. Invariants of this group are ential Galois group of the variational equations along related to first integrals of (48). We describe just main is Abelian. ingredients of this relation and its application to integrability study; for more details, see, e.g., books [62,75] In order to apply this theorem, it is necessary to use and papers [71,76]. effective methods which allow to determine properties We assume that entries of matrix M(t) in equa- of the differential Galois group of linear equations. In tion (48) are elements of a differential field F, e.g., most of the investigated systems, variational equations of rational functions F = C(t), or meromorphic func- contain a closed smaller s-th dimensional subsystem tions F = M(t). It is obvious that solutions of (47)are called normal variational equations and analysis of its 123 Integrability analysis of chaotic differential Galois group is sufficient. Moreover, fre- where we used assumption [u j , v]=0for1 ≤ quently normal variational equations can be rational- j ≤ n − k. Hence, F j+k are first integrals of sys- ized, i.e., transformed by means of change of indepen- tem (49). Using the fact that [u j , ui ]=0, one can dent variable t → f (x(t)),intos-th order differential also show that F1,...,Fn pairwise commute. Let equation with rational coefficients. In the case when ϕ(t) = (ϕ1(t), . . . , ϕn(t)) be a particular solution of s = 2, the abelianity of the identity component of the equations (47). Then differential Galois group can be investigated by means t −→ ψ(t):= (ϕ(t), 0) ∈ C2n (52) of the Kovacic algorithm [69]. After successful applications of Morales–Ramis the- is a particular solution of Hamiltonian system (50). The ory to various Hamiltonian systems, it was natural to variational equations along this particular solution are extend its application for non-Hamiltonian systems. As given by we already mentioned, for such systems, there is no X˙ = M(t)X, Y˙ =−M(t)T Y, commonly accepted definition of integrability. How- ∂v (53) ever, if we restrict ourselves to the B-integrability, where M(t) = (ϕ(t)). then we have an elegant generalization of this theory. ∂x Namely, with system (6) we can associate its cotan- Notice that first system of (53) is simply variational gent lift, i.e., a Hamiltonian system with the following equations (48), and the second one is just its adjoint. Hamiltonian function Hence, if is a fundamental matrix of the system ˙ −1 T n X = M(t)X, then := ( ) is a fundamental ˙ =− ( )T H = pi vi (x), (49) matrix of the equations Y M t Y.Itimpliesthat i=1 the differential Galois group of whole system (53) coincides with the differential Galois group of variational where x = (x1,...,xn) and p = (p1,...,pn) are canonical coordinates defined in a symplectic manifold equations (48). Based on this observation, Ayoul and M = C2n. Hence, Hamiltonian vector field is given by Zung proved the following theorem that gives necessary integrability condition for non-Hamiltonian sys- d ∂H tems. xi = = vi (x), dt ∂pi n Theorem 7 (Ayoul–Zung [66]) Assume that system d ∂H ∂vj pi =− =− p j (x), 1 ≤ i ≤ n. (50) (47) is meromorphically B-integrable, then the iden- dt ∂xi ∂xi j=1 tity component of the differential Galois group of variational equations (48) along a particular solution ϕ(t) In paper [66], Ayoul and Zung showed that if is Abelian. system (47)isB-integrable with k first integrals F ,...,F − ( ) = 1 k and commuting n k symmetries u1 x This theorem is the main tool to integrability study of v( ), ( ),..., ( ) x u2 x un−k x , then a lifted system gen- the non-Hamiltonian systems considered in this work. erated by function (49) is integrable in the Liouville sense. The above observation was nicely sketched in [71]. Namely, we start with defining the following func- References tions n 1. Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci. F ( , ) := , ( ):= j ( ), 20 j+k x p p u j x pi ui x , 130–141 (1963) i=1 2. May, R.M.: Simple mathematical models with very compli- 1 ≤ j ≤ n − k. cated dynamics. Nature 261, 459–467 (1976) 3. Hénon, M.: A two-dimensional mapping with a strange Calculation of the Poisson bracket of these functions attractor. Commun. Math. Phys. 50, 69–77 (1976) 4. Matsumoto, T.: A chaotic attractor from Chua’s circuit. IEEE with Hamiltonian (49)gives Trans. Circuits Syst. 31, 1055–1058 (1984) n 5. Chen, G.R., Ueta, T.: Yet another chaotic attractor. Int. J. ∂F j+k ∂H ∂F j+k ∂H Bifurc. Chaos 9, 1465–1466 (1999) {F + , H}= − j k 6. Lü, J.H., Chen, G.R., Cheng, D.Z., Celikovský,ˇ S.: Bridge ∂xi ∂pi ∂pi ∂xi i=1 the gap between the Lorenz system and the Chen system. =p, [u j , v] = 0, (51) Int. J. Bifurc. Chaos 12, 2917–2926 (2002) 123 W. Szuminski ´

7. Qi, G.Y.,Chen, G.R., Wyk, M.: A four-wing chaotic attractor hyper-chaotic multi-scroll attractors. Chaos Soliton Fract. generated from a new 3D quadratic autonomous system. 105, 244–255 (2017) Chaos Soliton Fract. 38, 705–721 (2008) 26. Rajagopal, K., Karthikeyan, A., Duraisamy, P.: Hyper- 8. Li, C.B., Wang, D.C.: An attractor with invariable Lyapunov chaotic chameleon: fractional order FPGA implementation. exponent spectrum and its Jerk circuit implementation. Acta Complexity, Article ID 8979408 (2017) https://doi.org/10. Phys. Sin. 58, 764–770 (2009) 1155/2017/8979408 9. Li, C., Li, H., Tong, Y.: Analysis of a novel three- 27. Singh, J.P., Roy, B.K., Jafari, S.: New family of 4-D hyper- dimensional chaotic system. Optik 124, 1516–1522 (2013) chaotic and chaotic systems with quadric surfaces of equi- 10. Tahir, F.R., Ali, R.S., Pham, V.-T., Buscarino, A., Frasca, libria. Chaos Soliton Fract. 106, 243–257 (2018) M., Fortuna, L.: A novel 4D autonomous 2n-butterfly wing 28. Hajipour, A., Hajipour, M., Baleanu, D.: On the adaptive chaotic attractor. Nonlinear Dyn. 85, 2665–2671 (2016) sliding mode controller for a hyperchaotic fractional-order 11. Jafari, M., Mliki, E., Akgul, A., Pham, V.-T., Kingni, S.T., financial system. Physica A 497, 139–153 (2018) Wang, X., Jafari, S.: Chameleon: the most hidden chaotic 29. May, R.M., Beddington, J.R.: Nonlinear difference equa- flow. Nonlinear Dyn. 88, 2303–2317 (2017) tions: stable points, stable cycles, chaos (1975) (Unpub- 12. Dadras, S., Momeni, H.R., Qi, G.: Analysis of a new 3D lished manuscript) smooth autonomous system with different wing chaotic 30. Baumol, W.J., Banhabib, J.: Chaos: significance, mecha- attractors and transient chaos. Nonlinear Dyn. 62, 391–405 nism, and economic applications. J. Econ. Perspect. 3(1), (2010) 77–105 (1989) 13. Wang, Z., Ma, J., Chen, Z., Zhang, Q.: A new chaotic system 31. Chian, A.C.-L.: Nonlinear dynamics and chaos in macroeco- with positive topological entropy. Entropy 17, 5561–5579 nomics. Int. J. Theor. Appl. Finance 63(3), 601–601 (2000) (2015) 32. Chian, A.C.-L., Borotto, F.A., Rempel, E.L., Rogers, C.: 14. Tong, Y.N.: Dynamics of a three-dimensional chaotic sys- Attractor merging crisis in chaotic business cycles. Chaos tem. Optik 126, 5563–5565 (2015) Soliton Fract. 24(3), 869–875 (2005) 15. Kengne, J., Folifack Signing, V.R., Chedjou, J.C., Leutcho, 33. Kaldor, N.: A model of economic growth. Econ. J. 67(268), G.D., Leutcho, G.D.: Nonlinear behavior of a novel chaotic 591–624 (1957) jerk system: antimonotonicity, crises, and multiple coexist- 34. Orlando, G.: A discrete mathematical model for chaotic ing attractors. Int. J. Dyn. Control (1957). https://doi.org/ dynamics in economics: Kaldor’s model on business cycle. 10.1007/s40435-017-0318-6 Math. Comput. Simul. 125, 83–98 (2016) 16. He, Y., Xu, H.M.: Yet another four-dimensional chaotic sys- 35. Hicks, J.R.: Mr. Keynes and the classics: a suggested inter- tem with multiple coexisting attractors. Optik 132, 124–31 pretation. Commun. Math. Phys. 5(2), 147–159 (1937) (2017) 36. De Cesare, L., Sportelli, M.: A dynamic IS-LM model with 17. Borah, M., Roy, B.K.: An enhanced multi-wing fractional- delayed taxation revenues. Chaos Soliton Fract. 25(1), 233– order chaotic system with coexisting attractors and switch- 244 (2005) ing hybrid synchronisation with its nonautonomous coun- 37. Fanti, L., Manfredi, P.: Chaotic business cycles and fiscal terpart. Chaos Soliton Fract. 102, 372–386 (2017) policy: an IS-LM model with distributed tax collection lags. 18. Borah, M., Roy, B.K.: Can fractional-order coexisting attrac- Chaos Soliton Fract. 32(2), 736–744 (2007) tors undergo a rotational phenomenon? ISA Trans. (2017). 38. Lorenz, H.W., Nusse, H.E.: Chaotic attractors, chaotic sad- https://doi.org/10.1016/j.isatra.2017.02.007 dles, and fractal basin boundaries: Goodwin’s nonlinear 19. Borah, M., Roy, B.K.: Dynamics of the fractional-order accelerator model reconsidered. Chaos Soliton Fract. 13(5), chaotic PMSG, its stabilisation using predictive control 957–965 (2002) and circuit validation. ET Electr. Power Appl. 11, 707–716 39. Matsumoto, A., Merlone, U., Szidarovszky, F.: Goodwin (2017) accelerator model revisited with fixed time delays. Com- 20. Rössler, O.E.: An equation for hyperchaos. Phys. Lett. A mun. Nonlinear Sci. Numer. Simul. 58, 233–248 (2018) 71(2–3), 55–157 (1979) 40. Gori, L., Guerrini, L., Sodini, M.: Disequilibrium dynamics 21. Mahmoud, G.M., Mahmoud, E.E., Ahmed, M.E.: On the in a Keynesian model with time delays. Commun. Nonlinear hyperchaotic complex luü system. Nonlinear Dyn. 58, 725– Sci. Numer. Simul. 58, 119–130 (2018) 738 (2009) 41. Iwaszczuk, N., Kavalet, I.: Delayed feedback control 22. Yu, H., Cai, G., Li, Y.: Dynamic analysis and control of method for generalized Cournot–Puu oligopoly model. In: a new hyperchaotic finance system. Nonlinear Dyn. 67(3), Iwaszczuk, N. (ed.) Selected Economic and Technological 2171–2182 (2012) Aspects of Management, pp. 108–123. Wydawnictwo AGH 23. Nigmatullin, R.R., Osokin, S.I., Awrejcewicz, J., Kudra, G.: w Krakowie, Kraków (2013) Application of the generalized Prony spectrum for extrac- 42. Huang, D., Li, H.: Theory and Method of the Nonlinear tion of information hidden in chaotic trajectories of triple Economics. Sichuan University Press, Chengdu (1993) pendulum. Central Eur. J. Phys. 12(8), 565–77 (2014) 43. Cai, G., Huang, J.: A new finance chaotic attractor. Int. J. 24. Kai, G., Zhang, W., Wei, Z.C., Wang, J.F., Akgul, A.: Hopf Nonlinear Sci. 3(3), 213–220 (2007) bifurcation, positively invariant set, and physical realization 44. Gardini, L., Kubin, I., Tramontana, F., Wagener, F.: Fore- of a new four-dimensional hyperchaotic financial system. word to the special issue on “Dynamic Models in Economics Math. Probl. Eng., Article ID 2490580 (2017) https://doi. and Finance”. Commun. Nonlinear Sci. Numer. Simul. 58, org/10.1155/2017/2490580 1–328 (2018) 25. Chen, L., Pan, W., Wang, K., Wu, R., Tenreiro Machado, 45. Ma, J., Chen, Y.: Study for the bifurcation topological struc- J.A., Lopes, A.M.: Generation of a family of fractional order ture and the global complicated character of a kind of non- 123 Integrability analysis of chaotic

linear finance system. II. Appl. Math. Mech. 22(12), 1236– 62. Morales-Ruiz, J.J.: Differential Galois Theory and Non- 1242 (2001) integrability of Hamiltonian Systems. Progress in Mathe- 46. Ma, J., Chen, Y.: Study for the bifurcation topological struc- matics. Birkhauser Verlag, Basel (1999) ture and the global complicated character of a kind of nonlin- 63. Pujol, O., Pérez, J.P., Ramis, J.P., Simó, C., Simon, S., Weil, ear finance system. I. Appl. Math. Mech. 22(11), 1119–1128 J.A.: Swinging Atwood machine: experimental and numeri- (2001) cal results, and a theoretical study. Physica D 239(12), 1067– 47. Gao, Q., Ma, J.: Chaos and Hopf bifurcation of a finance 1081 (2010) system. Nonlinear Dyn. 58(1–2), 209–216 (2009) 64. Szumiński, W., Maciejewski, A.J., Przybylska, M.: Note on 48. Zhao, X., Li, Z., Li, S.: Synchronization of a chaotic finance integrability of certain homogeneous Hamiltonian systems. system. Appl. Math. Comput. 217(13), 6031–6039 (2011) Phys. Lett. A 379(45–46), 2970–2976 (2015) 49. Zhou, W., Pan, L., Li, Z., Halang, W.A.: Non-linear feedback 65. Maciejewski, A.J., Przybylska, M.: Integrability of Hamilto- control of a novel chaotic system. Int. J. Control Autom. nian systems with algebraic potentials. Phys. Lett. A 380(1– Syst. 7(6), 939 (2009) 2), 76–82 (2016) 50. Jabbari, A., Kheiri, H.: Anti-synchronization of a modi- 66. Ayoul, M., Zung, N.T.: Galoisian obstructions to non- fied three-dimensional chaotic finance system with uncer- Hamiltonian integrability. C. R. Math. Acad. Sci. Paris tain parameters via adaptive control. Int. J. Nonlinear Sci. 348(23–24), 1323–1326 (2010) 14(2), 178–185 (2012) 67. Maciejewski, A.J., Przybylska, M.: Non-integrability of 51. Kocamaz, U.E., Göksu, A., Taskin, H., Uyaroglu, Y.: Syn- ABC flow. Phys. Lett. A 303(4), 265–272 (2002) chronization of chaos in nonlinear finance system by means 68. Przybylska, M.: Differential Galois obstructions for integra- of sliding mode and passive control methods: a comparative bility of homogeneous Newton equations. J. Math. Phys. 49, study. ITC 44, 172–181 (2015) 022701-1–022701-40 (2008) 52. Yang, M., Cai, G.: Chaos control of a non-linear finance 69. Kovacic, J.J.: An algorithm for solving second order linear system. J. Uncertain Syst. 5(4), 263–270 (2011) homogeneous differential equations. J. Symb. Comput. 2(1), 53. Chen, W.C.: Dynamics and control of a financial system 3–43 (1986) with time-delayed feedbacks. Chaos Soliton Fract. 37(4), 70. Van der Put, M., Singer, M.F.: Galois Theory of Linear Dif- 1198–1207 (2006) ferential Equations. Springer, Berlin (2003) 54. Chen, X., Liu, H., Xu, C.: The new result on delayed finance 71. Maciejewski, A.J., Przybylska, M.: Differential Galois the- system. Nonlinear Dyn. 78, 1989–1998 (2014) ory and integrability. Int. J. Geom. Methods Mod. Phys. 6(8), 55. Nosé, S.: Constant temperature molecular dynamics meth- 1357–1390 (2009) ods. Prog. Theor. Phys. Suppl. 103, 1–46 (1991) 72. Bogoyavlenski, O.I.: The A-B-C-cohomologies for dynam- 56. Hoover, W.G.: Remark on some simple chaotic flows. Phys. ical systems. C. R. Math. Rep. Acad. Sci. Canada 18(5), Rev. E 51, 159–760 (1995) 99–204 (1996) 57. Shivamoggi, B.K.: Nonlinear Dynamics and Chaotic Phe- 73. Bogoyavlenski, O.I.: A concept of integrability of dynamical nomena: An Introduction. Springer, Amsterdam (2014) systems. C. R. Math. Rep. Acad. Sci. Canada 18(5), 99–204 58. Alligood, K.T., Sauer, T.D., Yorke, J.A.: Chaos: An Intro- (1996) duction to Dynamical Systems. Springer, New York (1996) 74. Ziglin, S.L.: Branching of solutions and non-existence of 59. Wolf, A., Swift, J., Swinney, H., John, A.: Determining Lya- first integrals in Hamiltonian mechanics. I. Funct. Anal. punov exponents from a time series. Physica D 16, 285–317 Appl. 16, 181–189 (1982) (1985) 75. Audin, M.: Hamiltonian Systems and Their Integrability. 60. Kaplan, J.L., Yorke, J.A.: Chaotic behavior of multidi- AMS, Providence (2008) mensional difference equations. In: Functional Differential 76. Maciejewski, A.J., Przybylska, M.: Non-integrability of the Equations and Approximation of Fixed Points (Proc. Sum- problem of a rigid satellite in gravitational and magnetic mer School and Conf., Univ. Bonn, Bonn, 1978). Lecture fields. Celest. Mech. Dyn. Astron. 87(4), 317–351 (2003) Notes in Math., vol. 730, pp. 204–227. Springer, Berlin (1979) 61. Wu, W.J., Chen, Z.Q.: Hopf bifurcation and intermittent transition to hyperchaos in a novel strong four-dimensional hyperchaotic system. Nonlinear Dyn. 60, 615–630 (2010)

123