Lyapunov dimension formula for the attractor of the Glukhovsky–Dolzhansky system

Exact formulas for the Lyapunov dimension of attractors of the generalized Lorenz system and the Glukhovsky–Dolzhansky system are obtained.     

Lyapunov functions in the attractors dimension theory

The effectiveness of constructing Lyapunov functions in the attractors dimension theory is theory of the dimension demonstrated. Formulae for the Lyapunov dimension of the Lorenz, Hénon and Chirikov attractors are derived and proved. A hypothesis regarding the formula for the dimension of the Rössler attractor is formulated.     

Numerical justification of Leonov conjecture on Lyapunov dimension of Rossler attractor

Exact Lyapunov dimension of attractors of many classical chaotic systems (such as Lorenz, Henon, and Chirikov systems) is obtained. While exact Lyapunov dimension for Rössler system is not known, Leonov formulated the following conjecture: Lyapunov dimension of Rössler attractor is equal to local Lyapunov dimension in one of its stationary points. In the present work Leonov’s conjecture on Lyapunov dimension of various Rössler systems with standard parameters is checked numerically.     

Lyapunov Functions in Dimension Estimates of Attractors of Dynamical Systems

We discuss relations between the Lyapunov dimension and the topological, Hausdorff, and fractal dimensions of attractors. For the Henon and Lorenz systems, explicit formulas characterizing the Lyapunov dimension of their attractors are obtained. Bibliography: 23 titles.     

Lyapunov dimension formulas for Lorenz-like systems

A general formula for the Lyapunov dimension of attractors of the Lorenz and Tigan systems is derived in the case when all their equilibrium states are hyperbolic. Lyapunov dimension formulas are obtained for the classical Lorentz parameters σ > 0, r > 1, and b ∈ (0, 4]. The problems of deriving analytical dimension formulas for the Chen and Lu systems and for the global stability of the Lorenz system are formulated.     

Qualitative behavior of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres

In this article, the bounds of the Lorenz‐like chaotic system describing the flow between two concentric rotating spheres have been studied. Based on Lagrange multiplier method, the function extremum theory and the generalized positive definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and the globally exponentially attractive set for this system. The results that obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension and the Lyapunov dimension of chaotic attractors. © 2016 Wiley Periodicals, Inc. Complexity 21: 67–72, 2016     

Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors

This paper surveys results of the authors and others conceming estimates for the Hausdorff dimension of strange attractors, particularly in the case of (generalized) Lorenz systems and Rössler systems. A key idea is the interpretation of Hausdorff measure as an analogue of a Lyapunov function.     

When Darwin meets Lorenz: Evolving new chaotic attractors through genetic programming

In this paper, we propose a novel methodology for automatically finding new chaotic attractors through a computational intelligence technique known as multi-gene genetic programming (MGGP). We apply this technique to the case of the Lorenz attractor and evolve several new chaotic attractors based on the basic Lorenz template. The MGGP algorithm automatically finds new nonlinear expressions for the different state variables starting from the original Lorenz system. The Lyapunov exponents of each of the attractors are calculated numerically based on the time series of the state variables using time delay embedding techniques. The MGGP algorithm tries to search the functional space of the attractors by aiming to maximise the largest Lyapunov exponent (LLE) of the evolved attractors. To demonstrate the potential of the proposed methodology, we report over one hundred new chaotic attractor structures along with their parameters, which are evolved from just the Lorenz system alone.     

Generating a new chaotic attractor by feedback controlling method

A new chaotic system is found by feedback controlling method in this paper. According to the definition of the generalized Lorenz system, the new chaotic system does not belong to generalized Lorenz systems. We analyze the new system by means of phase portraits, Lyapunov exponents, fractional dimension, bifurcation diagram, and Poincaré map. The particular interest is that this novel system can generate two one‐scroll and one two‐scroll chaotic attractors with the variation of a single parameter. The obtained results show clearly that the system is a new chaotic system and deserves a further detailed investigation. Copyright © 2011 John Wiley & Sons, Ltd.     

Chaos in fractional conjugate Lorenz system and its scaling attractors

Chaotic dynamics of fractional conjugate Lorenz system are demonstrated in terms of local stability and largest Lyapunov exponent. Chaos does exist in the fractional conjugate Lorenz system with order less than three since it has positive largest Lyapunov exponent. Furthermore, scaling chaotic attractors of fractional conjugate Lorenz system is theoretically and numerically analyzed with the help of one-way synchronization method and adaptive synchronization method. Numerical simulations are performed to verify the theoretical analysis.     

Analytical and numerical investigation of two families of Lorenz-like dynamical systems

We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke–Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to  = ±1, m = 0, 1.

(I)

The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal (t → ∞) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters σ, r, b, ω and m.     

Evolving chaos: Identifying new attractors of the generalised Lorenz family

In a recent paper, we presented an intelligent evolutionary search technique through genetic programming (GP) for finding new analytical expressions of nonlinear dynamical systems, similar to the classical Lorenz attractor's which also exhibit chaotic behaviour in the phase space. In this paper, we extend our previous finding to explore yet another gallery of new chaotic attractors which are derived from the original Lorenz system of equations. Compared to the previous exploration with sinusoidal type transcendental nonlinearity, here we focus on only cross-product and higher-power type nonlinearities in the three state equations. We here report over 150 different structures of chaotic attractors along with their one set of parameter values, phase space dynamics and the Largest Lyapunov Exponents (LLE). The expressions of these new Lorenz-like nonlinear dynamical systems have been automatically evolved through multi-gene genetic programming (MGGP). In the past two decades, there have been many claims of designing new chaotic attractors as an incremental extension of the Lorenz family. We provide here a large family of chaotic systems whose structure closely resemble the original Lorenz system but with drastically different phase space dynamics. This advances the state of the art knowledge of discovering new chaotic systems which can find application in many real-world problems. This work may also find its archival value in future in the domain of new chaotic system discovery.     

Dynamical analysis of a new autonomous 3-D chaotic system only with stable equilibria

This paper presents a new 3-D autonomous chaotic system, which is topologically non-equivalent to the original Lorenz and all Lorenz-like systems. Of particular interest is that the chaotic system can generate double-scroll chaotic attractors in a very wide parameter domain with only two stable equilibria. The existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear. Finally, the complicated dynamics are studied by virtue of theoretical analysis, numerical simulation and Lyapunov exponents spectrum. The obtained results clearly show that the chaotic system deserves further detailed investigation.     

A note on multistep methods and attracting sets of dynamical systems

Summary We consider a dynamical system described by an autonomous ODE with an asymptotically stable attractor, a compact set of orbitrary shape, for which the stability can be characterized by a Lyapunov function. Using recent results of Eirola and Nevanlinna [1], we establish a uniform estimate for the change in value of this Lyapunov function on discrete trajectories of a consistent, strictly stable multistep method approximating the dynamical system. This estimate can then be used to determine nearby attracting sets and attractors for the discretized system as done in Kloeden and Lorenz [3, 4] for 1-step methods.This work was supported by the U.S. Department of Energy Contract DE-A503-76 ER72012     

A new butterfly-shaped attractor of Lorenz-like system

In this letter a new butterfly-shaped chaotic attractor is reported. Some basic dynamical properties, such as Poincare mapping, Lyapunov exponents, fractal dimension, continuous spectrum and chaotic dynamical behaviors of the new chaotic system are studied. Furthermore, we clarify that the chaotic attractors of the system is a compound structure obtained by merging together two simple attractors through a mirror operation.     

Features of the behavior of solutions to a nonlinear dynamical system in the case of two-frequency parametric resonance

Two-frequency parametric resonance in nonlinear dynamical systems is studied by analyzing a delay differential equation with the delay obeying a two-frequency law, which arises in the mathematical simulation of some physical processes. It is shown that the system can exhibit chaotic oscillations (strange attractors) when the parametric excitation frequencies are both close to the doubled eigenfrequency of the system (degenerate case). The formation mechanisms of chaotic attractors are discussed, and the Lyapunov exponents and the Lyapunov dimension are calculated for them. If only one of the parametric excitation frequencies is close to the double eigenfrequency, a two-frequency regime occurs in the system.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

Global attractors for porous-elasticity system from second spectrum viewpoint

This paper is concerned with the global attractors for a new partially damped porous-elasticity system by taking a truncated version which is free of blow-up on second wave speed. We establish the global well-posedness of the system via Faedo–Galerkin method. By considering only one damping term acting on the volume fraction in the system we prove the existence of absorbing set for the solution semigroup regardless any relationship between coefficients of the system. Finally, by using Lyapunov and recent quasi-stability methods we prove the existence of smooth global attractors with finite fractal dimension.     

Dynamical analysis and numerical simulation of bloch chaotic system

In this article, some dynamics of Bloch chaotic system have been studied. Based on Lagrange multiplier method, optimization theory, and the generalized positively definite and radially unbound Lyapunov functions with respect to the parameters of the system, we derive the ultimate bound and a family of mathematical expressions of globally exponentially attractive sets for this system with respect to the parameters of system. The results obtained in this article provides theory basis for chaotic synchronization, chaotic control, Hausdorff dimension, and Lyapunov dimension of chaotic attractors of Bloch chaotic system. © 2016 Wiley Periodicals, Inc. Complexity 21: 201–206, 2016     

A three-dimensional nonlinear system with a single heteroclinic trajectory

The study for singular trajectories of three-dimensional (3D) nonlinear systems is one of recent main interests. To the best of our knowledge, among the study for most of Lorenz or Lorenz-like systems, a pair of symmetric heteroclinic trajectories is always found due to the symmetry of those systems. Whether or not does there exist a 3D system that possesses a single heteroclinic trajectory? In the present note, based on a known Lorenz-type system, we introduce such a 3D nonlinear system with two cubic terms and one quadratic term to possess a single heteroclinic trajectory. To show its characters, we respectively use the center manifold theory, bifurcation theory, Lyapunov function and so on, to systematically analyse its complex dynamics, mainly for the distribution of its equilibrium points, the local stability, the expression of locally unstable manifold, the Hopf bifurcation, the invariant algebraic surface, and its homoclinic and heteroclinic trajectories, etc. One of the major results of this work is to rigorously prove that the proposed system has a single heteroclinic trajectory under some certain parameters. This kind of interesting phenomenon has not been previously reported in the Lorenz system family (because the huge amount of related research work always presents a pair of heteroclinic trajectories due to the symmetry of studied systems). What"s more key, not like most of Lorenz-type or Lorenz-like systems with singularly degenerate heteroclinic cycles and chaotic attractors, the new proposed system has neither singularly degenerate heteroclinic cycles nor chaotic attractors observed. Thus, this work represents an enriching contribution to the understanding of the dynamics of Lorenz attractor.