Two theorems of generalized unsynchronization for coupled chaotic systems

Generalized chaos synchronization has been widely studied and many control methods have been presented, but up to now no criterion has been given for generalized unsynchronization. The generalized unsynchronization means that the state variables of two coupled chaotic systems cannot approach generalized synchronization. In this paper, we propose two theorems which give the criteria of generalized unsynchronization for two different chaotic dynamic systems with whatever large strength of linear coupling. Two simulated examples are also given.     

An optimization algorithm based on chaotic behavior and fractal nature

In this paper, we propose a new optimization technique by modifying a chaos optimization algorithm (COA) based on the fractal theory. We first implement the weighted gradient direction-based chaos optimization in which the chaotic property is used to determine the initial choice of the optimization parameters both in the starting step and in the mutations applied when a convergence to local minima occurred. The algorithm is then improved by introducing a method to determine the optimal step size. This method is based on the fact that the sensitive dependence on the initial condition of a root finding technique (such as the Newton–Raphson search technique) has a fractal nature. From all roots (step sizes) found by the implemented technique, the one that most minimizes the cost function is employed in each iteration. Numerical simulation results are presented to evaluate the performance of the proposed algorithm.     

Generalized synchronization in linearly coupled time periodic systems

We consider the synchronization of a network of linearly coupled and not necessarily identical oscillators. We present an approach to the existence of the synchronization manifold which is based on some results developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that they are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski's principle. Several examples of application are presented.     

Heteroclinic chaotic behavior driven by a Brownian motion

In this paper we study the chaotic behavior of a planar ordinary differential system with a heteroclinic loop driven by a Brownian motion, an unbounded random forcing. Unlike the case of homoclinic loops, two random Melnikov functions are needed in order to investigate the intersection of stable segments of one saddle and unstable segments of the other saddle. We prove that for almost all paths of the Brownian motion the forced system admits a topological horseshoe of infinitely many branches. We apply this result to the Josephson junction and the soft spring Duffing oscillator.     

On the behavior of the Lorenz equation backward in time

The sets of solutions to the Lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined. The set associated with the middle eigenvalue is shown to project surjectively onto a plane, thereby providing a lower estimate for its dimension. Specific bounds are also found for a cone containing this set.     

Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré-Birkhoff theorem

In the general setting of a planar first order system

(0.1)     

Sell’s conjecture for non-autonomous dynamical systems

This paper is dedicated to the study of the G. Sell’s conjecture for general non-autonomous dynamical systems. We give a positive answer for this conjecture and we apply this result to different classes of non-autonomous evolution equations: Ordinary Differential Equations, Functional Differential Equations and Semi-linear Parabolic Equations.     

Carrying simplices in nonautonomous and random competitive Kolmogorov systems

The purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive Kolmogorov systems via the skew-product flows approach. It is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew-product flow associated with a nonautonomous (random) competitive Kolmogorov system.     

Invariant manifolds around equilibria of Newtonian equations: Some pathological examples

Let the equation be periodic in time, and let the equilibrium x_∗≡0 be a periodic minimizer. If it is hyperbolic, then the set of asymptotic solutions is a smooth curve in the plane ; this is stated by the Stable Manifold Theorem. The result can be extended to nonhyperbolic minimizers provided only that they are isolated and the equation is analytic (Ureña, 2007 [6]). In this paper we provide an example showing that one cannot say the same for C² equations. Our example is pathological both in a global sense (the global stable manifold is not arcwise connected), and in a local sense (the local stable manifolds are not locally connected and have points which are not accessible from the exterior).     

Affine embeddings and intersections of Cantor sets

Let E,F⊂R^d$E,F\subset {\mathbb{R}}^d$ be two self-similar sets. Under mild conditions, we show that F   can be C¹$C^1$-embedded into E if and only if it can be affinely embedded into E; furthermore if F cannot be affinely embedded into E  , then the Hausdorff dimension of the intersection E∩f(F)$E\cap f(F)$ is strictly less than that of F   for any C¹$C^1$-diffeomorphism f   on R^d${\mathbb{R}}^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of E and F if F can be affinely embedded into E  . As an application, we show that _dimH?E∩f(F)<min?{_dimH?E,_dimH?F}${\dim }_H?E\cap f(F)<\min ?\{{\dim }_H?E,{\dim }_H?F\}$ when E is any Cantor-p set and F any Cantor-q   set, where p,q?2$p,q?2$ are two integers with log?p/log?q∉Q$\log ?p/\log ?q\notin \mathbb{Q}$. This is related to a conjecture of Furstenberg about the intersections of Cantor sets.     

Periodic orbits for perturbations of piecewise linear systems

We consider the existence of periodic orbits in a class of three-dimensional piecewise linear systems. Firstly, we describe the dynamical behavior of a non-generic piecewise linear system which has two equilibria and one two-dimensional invariant manifold foliated by periodic orbits. The aim of this work is to study the periodic orbits of the continuum that persist under a piecewise linear perturbation of the system. In order to analyze this situation, we build a real function of real variable whose zeros are related to the limit cycles that remain after the perturbation. By using this function, we state some results of existence and stability of limit cycles in the perturbed system, as well as results of bifurcations of limit cycles. The techniques presented are similar to the Melnikov theory for smooth systems and the method of averaging.     

Optimal estimates on rotation number of almost periodic systems

In this paper, we will give some optimal estimates on the rotation number of the linear equation and that of the asymmetric equation: where p(t) and q(t) are almost periodic functions and These estimates are obtained by introducing some kind of new norms in the spaces of almost periodic functions. Supported by the National Natural Science Foundation of China (no. 10325102), TRAPOYT-M.O.E. of China (2001), and the National 973 Project of China (no. G1999075108). Received: April 6, 2004; revised: July 7, 2004     

Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems

The purpose of this paper is two-fold. Firstly, we will give some parabolic-like conditions which improve the well-known angle conditions and allow further computations of the critical groups both at degenerate critical points and at infinity. As an application, we then consider the second-order Hamiltonian systems

     

Homoclinic solutions for a class of non-autonomous subquadratic second-order Hamiltonian systems

In this paper we consider the existence of homoclinic solutions for the following second-order non-autonomous Hamiltonian system:

(HS)     

Discrete chaos induced by heteroclinic cycles connecting repellers in Banach spaces

This paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in Banach spaces. Several criteria of chaos are established in general Banach spaces and finite-dimensional spaces, respectively, by employing the coupled-expansion theory. All the maps presented in this paper are proved to be chaotic in the sense of both Li-Yorke and Devaney or in the sense of both Li-Yorke and Wiggins or in the sense of Li-Yorke. An illustrative example is provided with computer simulations.     

Positive periodic solutions of singular systems with a parameter

The existence and multiplicity of positive periodic solutions for second-order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our results provide a unified treatment for the problem and significantly improve several results in the literature. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.     

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor-repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.     

Reversible periodic orbits in a class of 3D continuous piecewise linear systems of differential equations

The so-called noose bifurcation is an interesting structure of reversible periodic orbits that was numerically detected by Kent and Elgin in the well-known Michelson system. In this work we perform an analysis of the periodic behavior of a piecewise version of the Michelson system where this bifurcation also exists. This variant is a one-parameter three-dimensional piecewise linear continuous system with two zones separated by a plane and it is also a representative of a wide class of reversible divergence-free systems.     

Smoothness of the carrying simplex for discrete-time competitive dynamical systems: A characterization of neat embedding

The paper is concerned with the question of smoothness of the carrying simplex S for a discrete-time dissipative competitive dynamical system. We give a necessary and sufficient criterion for S being a C¹ submanifold-with-corners neatly embedded in the nonnegative orthant, formulated in terms of inequalities between Lyapunov exponents for ergodic measures supported on the boundary of the orthant. This completes one thread of investigation occasioned by a question posed by M.W. Hirsch in 1988. Besides, amenable conditions are presented to guarantee the Cr (r?1) smoothness of S in the time-periodic competitive Kolmogorov systems of ODEs. Examples are also presented, one in which S is of class C¹ but not neatly embedded, the other in which S is not of class C¹.