Chaos–hyperchaos transition in coupled Rössler systems

doi:10.1016/S0375-9601(01)00651-X

Physics Letters A

Volume 290, Issues 3–4, 12 November 2001, Pages 139-144

https://doi.org/10.1016/S0375-9601(01)00651-X Get rights and content

Abstract

Many of the nonlinear high-dimensional systems have hyperchaotic attractors. Typical trajectory on such attractors is characterized by at least two positive Lyapunov exponents. We provide numerical evidence that chaos–hyperchaos transition in six-dimensional dynamical system given by flow can be characterized by the set of infinite number of unstable periodic orbits embedded in the attractor as it was previously shown for the case of two coupled discrete maps.

Introduction

Unstable periodic orbits (UPO's) constitute the most fundamental blocks of a chaotic system [1]. Theoretically, the infinite number of UPO's embedded in a chaotic set provides the skeleton of the attractor and allows the estimation of many dynamical invariants such as the natural measure, the spectra of Lyapunov exponents, the fractal dimension in the fundamental way [2]. Recently, UPO's have been used in the description of higher-dimensional dynamical phenomena such as blowout bifurcation [3] and chaos–hyperchaos transition (i.e., transition from the attractor characterized by one positive Lyapunov exponent to the attractor characterized by at least two positive exponents) [4]. It has been shown that chaos–hyperchaos transition as well as blowout bifurcation is mediated by an infinite number of UPO's which become repellers in the neighborhood of the transition point. The simultaneous existence of UPO's with different number of unstable direction gives rise to the nonhyperbolicity known as unstable dimension variability and provides a possible dynamic mechanism for the smooth transition through zero of second Lyapunov exponent.

Up to now, the description of chaos–hyperchaos transition using UPO's has been performed only for the case of coupled discrete maps. In this Letter, we argue and provide numerical evidence that this description can be applied to the continuous dynamical systems (flows). We show that the balance of the appropriate weights of UPO's orbits with one unstable dimension and UPO's with at least two unstable dimension gives the approximation of chaos–hyperchaos transition point.

Section snippets

The model

As an example consider two identical symmetrically coupled Rössler systems $x ̇_{1} =−x_{2} −x_{3}, x ̇_{2} =x_{1} +ax_{2}, x ̇_{3} =b+x_{3} (x_{1} −c)+d(y_{3} −x_{3}),$ $y ̇_{1} =−y_{2} −y_{3}, y ̇_{2} =y_{1} +ay_{2}, y ̇_{3} =b+y_{3} (y_{1} −c)+d(y_{3} −x_{3}),$ where $(x_{1},x_{2},x_{3},y_{1},y_{2},y_{3})∈ R^{6}$ are dynamical variables, a,b,c are system parameters and d is the coefficient of coupling. It is well known that the Rössler system develops continuous chaos through period-doubling bifurcation cascade [6]. Since the Rössler system has a foundation in the kinematics of chemical reaction [7], it is

Stability of low-periodic orbits embedded into the attractor

In the following, we try to investigate stability of low-periodic orbits embedded into the attractor of system (1) when it undergoes chaos–hyperchaos transition. In order to find and classify these orbits we use the Poincaré cross-section that is determined by the following normal vector $n =(−3.75,1.84,−6.48,1.75,−2.09,−0.10)$ and a base point with coordinates P=(1.72,0.35,3.40,−1.27,−2.29,0.53). Vector $n$ was chosen along flow (1) at point P. Fig. 3 shows the image of the chaotic attractor in

Conclusions

We have shown here that the transition from chaos to hyperchaos in higher-dimensional dynamical system given by a flow is a bifurcation that like in the case of the coupled maps, is mediated by an infinite number of unstable periodic orbits. In the neighborhood of the transition point one observes the co-existence of UPO's with one (saddles which are typical for 3-dimensional chaotic systems) and at least two unstable eigenvalues. This co-existence is responsible for the occurrence of

References (13)

C. Grebogi et al.
Phys. Rev. A
(1988)
D. Auerbach et al.
Phys. Rev. Lett.
(1987)
G.H. Gunaratne et al.
Phys. Rev. Lett.
(1987)
P. Cvitanovic
Chaos
(1992)
C. Grebogi et al.
Phys. Rev. A
(1988)
Y.C. Lai et al.
Phys. Rev. Lett.
(1997)
Y. Nagai et al.
Phys. Rev. E
(1997)
Y. Nagai et al.
Phys. Rev. E
(1997)
Y.-C. Lai
Phys. Rev. E
(1999)
T. Kapitaniak et al.
Phys. Rev. E
(2000)
T.S. Parker et al.
Practical Numerical Algorithms for Chaotic Systems
(1989)

There are more references available in the full text version of this article.

Cited by (64)

Emergence and evolution of unusual inhomogeneous limit cycles displacing hyperchaos in three quorum-sensing coupled identical ring oscillators
2023, Physica D: Nonlinear Phenomena
We demonstrate that strongly asymmetric limit cycles can be observed in the system of three identical ring oscillators (3-gene networks known as Repressilators) globally coupled by signal molecule diffusion added to the model in a way like the known bacterial “quorum-sensing” mechanism. These cycles are stable over a wide interval of the coupling strengths where they expel the dominant hyperchaotic regime existing in three Repressilators in very large areas of parameters. The bifurcations of the inhomogeneous limit cycle, with a high-amplitude orbit for one oscillator and two low-amplitude identical orbits for the other two, are traced. Bifurcation analysis reveals an unusual cascade of bifurcations ended in the appearance of a new limit cycle with splitted (slightly nonidentical) low-amplitude orbits. Both cycles lose stability giving birth inhomogeneous chaos in the small parameter interval. Hyperchaos dominates in the parameter plane around the “island” with inhomogeneous limit cycles, and this accounts for very long hyper chaotic transients when the system is returning to stable asymmetric cycles after their perturbations. In turn, it is the cycles that contribute the asymmetric and often rather long pieces in hyperchaotic trajectories. The presented cycles differ from the known asymmetric attractors: inhomogeneous limit cycles born from “oscillation death” and the cycles observing in “smallest chimeras”.
A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it's fractional order model and chaos synchronization using optimized fractional order sliding mode control
2021, Chaos, Solitons and Fractals
Citation Excerpt :
Many fractional order chaotic and hyper chaotic systems are generated and investigated from classical ones. As typical examples, like Lorenz [46–48], like Röslor [49–52], like Chua [53] and like Chen [54] and like Lü [55]. Future more, new and effective methods for the time-domain analysis of fractional-order dynamical systems are required for solving problems of control theory.
This research article reports the finding of a new twelve term hyper-chaotic system with three nonlinearities in a fourth dimensional order, obtained by adding a nonlinear state feedback to the first equation and a cross-product nonlinear term to the third equation of the 3 dimensional famous Rösslor continuous-time chaotic system with two new optimized additive control parameters. Firstly various basic dynamic properties have been analyzed. Then the stabilization of the new hyper-chaotic motion trajectories via genetically optimized linear feedback controllers to zero, as the direct Lyapunov stability analysis of the robust controller and the Lipschitz condition to get an appropriate negative define Lyapunov function which contain non linear third order terms is difficult, the stability of the resulting error system in a given finite time is proved. The fractional order model and it’s synchronization are also included, based on an optimized fractional order sliding mode controller, not solved analytically but using genetic algorithm which can give a smart solution with an objective function and some additive unknown parameters. Numerical simulations of phase portraits (integer and fractional), bifurcation diagrams and spectrum of Lyapunov exponents are presented, chaos stabilization and synchronization are depicted and show the use of controllers scheme.
Synchronized chaotic swinging of parametrically driven pendulums
2020, International Journal of Mechanical Sciences
Citation Excerpt :
Independently of N, the synchronized chaotic swinging occurs in the same interval of the driving frequency (0.941<η <0.966, see Fig. 4(c)), where the largest LE of the system is positive (λ1 > 0) and the second LE, being the largest CLE, fulfills the condition λ2 = λC ≤ 0. Increase of η over the threshold of negative λC (η >0.966) leads to synchrony loss, which is accompanied by the chaos–hyperchaos transition [46]. An example of a time graph, showing the convergence to zero of the synchronization error for the case of one hundred pendulums, is presented in Fig. 6(a).
We study the problem of complete synchronization of a pendulums' array attached to a common, externally forced structure. The presented analysis is focused on chaotic synchronization of the swinging, parametric pendulums. We propose a simplified criterion of synchronization for an arbitrary number of parametrically driven oscillators, which is based on the idea of a so-called autonomous driver decomposition and conditional Lyapunov exponents. According to this proposal, complete synchronization of any number of parametric oscillators driven via a common transmitter of an external signal (the structure) can be analyzed using a three degrees of freedom system in a master-slave configuration. The reduced system consists of an equivalent synchronous oscillator coupled with its virtual replica via the linking structure. It is shown that introduction of the intermediate structure allows chaotic synchronization of parametric pendulums, even in the presence of parameter's mismatch. The idea of an intermediate transmitter and the methodology of verification of stability of the synchronous state can be applied for any set of dynamical systems with a common external drive.
Chaotic synchronization in a pair of pendulums attached to driven structure
2018, International Journal of Non-Linear Mechanics
Citation Excerpt :
Moreover, these exponents are valuable criterion for assessing the stability of synchronous state. In the context of synchronization analysis, concepts of transversal [15,39], conditional [13,14,17] or response [18] Lyapunov exponents were introduced. The synchronous state is stable if these exponents are negative.
In this paper a mechanical system with three degrees of freedom, composed of two pendulums attached to a common, parametrically excited mass, is investigated. The presented analysis is focused on the complete synchronization of the swinging pendulums. We show that application of such structure of the system allows chaotic synchronization of pendulums. In the presence of parameters mismatch, chaotic synchrony is manifested by continual transitions between in-phase and anti-phase oscillations.
Infinite lattice of hyperchaotic strange attractors
2018, Chaos, Solitons and Fractals
By introducing trigonometric functions in a 4-D hyperchaotic snap system, infinite 1-D, 2-D, and 3-D lattices of hyperchaotic strange attractors were produced. Furthermore a general approach was developed for constructing self-reproducing systems, in which infinitely many attractors share the same Lyapunov exponents. In this case, cumbersome constants are necessary to obtain offset boosting; correspondingly additional periodic functions are needed for attractor hatching. As an example, a hyperchaotic system with a hidden attractor was transformed for reproducing 1-D, 2-D infinite lattices of hyperchaotic attractors and a 4-D lattice of chaotic attractors.
Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol oscillators
2017, Chinese Journal of Physics
Bifurcations and chaos in two-coupled periodically driven four-well Duffing-van der Pol (DVP) oscillators are numerically investigated as a function of the strength (δ) of nonlinear coupling and the amplitude (f) of the periodic driving force. The influence of weak coupling (δ ≪ 1) and strong coupling (δ ≫ 1) is analyzed with and without the external periodic force. For strong coupling (δ ≫ 1), the system shows completely chaotic behaviour but for weak coupling the system exhibits period-doubling and period-tripling routes to chaos, quasiperiodic orbit, reverse period-doubling bifurcation, periodic windows, chaotic orbit and intermittent behaviours for specific set of values of the parameters. The effect of phase shift (ϕ) of the system is also analyzed. Numerical results are demonstrated by constructing the bifurcation diagram, phase portrait and Poincar $\overset{´}{e}$ map.

View all citing articles on Scopus

View full text

Chaos–hyperchaos transition in coupled Rössler systems

Abstract

Introduction

Section snippets

The model

Stability of low-periodic orbits embedded into the attractor

Conclusions

Phys. Rev. A

Phys. Rev. Lett.

Phys. Rev. Lett.

Chaos

Phys. Rev. A

Phys. Rev. Lett.

Phys. Rev. E

Phys. Rev. E

Phys. Rev. E

Phys. Rev. E

Practical Numerical Algorithms for Chaotic Systems