Snap-back Repeller Research Articles

Periodicity and convergence for xn+1 =| xn− xn−1 |

Each solution { x n } of the equation in the title is either eventually periodic with period 3 or else, it converges to zero—which case occurs depends on whether the ratio of the initial values of { x n } is rational or irrational. Further, the sequence of ratios { x n / x n−1 } satisfies a first-order difference equation that has periodic orbits of all integer periods except 3. p-cycles for each p≠3 are explicitly determined in terms of the Fibonacci numbers. In spite of the non-existence of period 3, the unique positive fixed point of the first-order equation is shown to be a snap-back repeller so the irrational ratios behave chaotically.

The Marotto Theorem on planar monotone or competitive maps

In 1978, Marotto generalized Li–Yorke’s results on the criterion for chaos from one-dimensional discrete dynamical systems to n-dimensional discrete dynamical systems, showing that the existence of a non-degenerate snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is very useful in predicting and analyzing discrete chaos in multi-dimensional dynamical systems. Yet, besides it is well known that there exists an error in the conditions of the original Marotto Theorem, and several authors had tried to correct it in different way, Chen, Hsu and Zhou pointed out that the verification of “non-degeneracy” of a snap-back repeller is the most difficult in general and expected, “almost beyond reasonable doubt,” that the existence of only degenerate snap-back repeller still implies chaotic, which was posed as a conjecture by them. In this paper, we shall give necessary and sufficient conditions of chaos in the sense of Li–Yorke for planar monotone or competitive discrete dynamical systems and solve Chen–Hsu–Zhou Conjecture for such kinds of systems.

An improved version of the Marotto Theorem

In 1975, Li and Yorke introduced the first precise definition of discrete chaos and established a very simple criterion for chaos in one-dimensional difference equations, “period three implies chaos” for brevity. After three years. Marotto generalized this result to n-dimensional difference equations, showing that the existence of a snap-back repeller implies chaos in the sense of Li–Yorke. This theorem is up to now the best one in predicting and analyzing discrete chaos in multidimensional difference equations. Yet, it is well known that there exists an error in the condition of the original Marotto Theorem, and several authors had tried to correct it in different ways. In this paper, we further clarify the issue, with an improved version of the Marotto Theorem derived.

Transversal homoclinic orbits in a transiently chaotic neural network.

We study the existence of snap-back repellers, hence the existence of transversal homoclinic orbits in a discrete-time neural network. Chaotic behaviors for the network system in the sense of Li and Yorke or Marotto can then be concluded. The result is established by analyzing the structures of the system and allocating suitable parameters in constructing the fixed points and their pre-images for the system. The investigation provides a theoretical confirmation on the scenario of transient chaos for the system. All the parameter conditions for the theory can be examined numerically. The numerical ranges for the parameters which yield chaotic dynamics and convergent dynamics provide significant information in the annealing process in solving combinatorial optimization problems using this transiently chaotic neural network. (c) 2002 American Institute of Physics.

Chaotic Invariant Sets of High-Dimensional Hénon-Like Maps

High-dimensional Hénon-like maps have many applications in the research of spatial chaos and traveling waves of extended systems. Meanwhile, they are of great interest in their own right. The aim of this paper is, by applying the implicit function theorem, to show for high-dimensional Hénon-like maps the existence of chaotic invariant sets and the density of homoclinic points and heteroclinic points in them. Our method is motivated by Aubry's “anti-integrability” concept and is rather different from the traditional techniques such as horseshoes, transversal homoclinic points and heteroclinic cycles, and snap-back repellers.

TOPOLOGICAL STRUCTURE OF CHAOS IN DISCRETE-TIME NEURAL NETWORKS WITH GENERALIZED INPUT–OUTPUT FUNCTION

By analogue of [Chen & Aihara, 1995, 1997, 1999], we theoretically investigate the topologically chaotic structure, attracting set and global searching ability of discrete-time recurrent neural networks with the form of [Formula: see text] where the input–output function is defined as a generalized sigmoid function, such as vi= tanh (μiui), [Formula: see text] and [Formula: see text], etc. We first derive sufficient conditions of existence for a fixed point, and then prove that this fixed point eventually evolves into a snap-back repeller which generates chaotic structure when certain conditions are satisfied. Furthermore we prove that there exists an attracting set which includes not only the homoclinic orbit but also the globally unstable set of fixed points, thereby ensuring the neural networks to have global searching ability.Numerical simulations are also provided to demonstrate the theoretical results. The results indicated in this paper can be viewed as an extension of the works of [Chen & Aihara, 1997, 1999] and others [Gopalsamy & He, 1994; Wang & Smith, 1998].

Snap-back repellers and scrambled sets in general topological spaces

Snap-back repellers and scrambled sets in general topological spaces

Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the span

A wave equation on a one-dimensional interval I has a van der Pol type nonlinear boundary condition at the right end. At the left end, the boundary condition is fixed. At exactly the midpoint of the interval I, energy is injected into the system through a pair of transmission conditions in the feedback form of anti-damping. We wish to study chaotic wave propagation in the system. A cause of chaos by snapback repellers has been identified. These snapback repellers are repelling fixed points possessing homoclinic orbits of the non-invertible map in 2D corresponding to wave reflections and transmissions at, respectively, the boundary and the middle-of-the-span points. Existing literature [F. R. Marotto, J. Math. Anal. Appl. 63, 199–223 (1978)] on snapback repellers contains an error. We clarify the error and give a refined theorem that snapback repellers imply chaos. Numerical simulations of chaotic vibration are also illustrated.

Chaos and asymptotical stability in discrete-time neural networks

This paper aims to theoretically prove by applying Marotto's Theorem that both transiently chaotic neural networks (TCNN) and discrete-time recurrent neural networks (DRNN) have chaotic structure. A significant property TCNN and DRNN is that they have only one bounded fixed point, when absolute values of the self-feedback connection weights in TCNN and the difference time in DRNN are sufficiently large. We show that this unique fixed point can actually evolve into a snap-back repeller which generates chaotic structure, if several conditions are satisfied. On the other hand, by using the Lyapunov functions, we also derive sufficient conditions on asymptotical stability for symmetrical versions of both TCNN and DRNN, under which TCNN and DRNN asymptotically converge to a fixed point. Furthermore, related bifurcations are also considered in this paper. Since both TCNN and DRNN are not special but simple and general, the obtained theoretical results hold for a wide class of discrete-time neural networks. To demonstrate the theoretical results of this paper better, several numerical simulations are provided as illustrating examples.

CHAOTIC RAMSEY EQUILIBRIUM

We consider an economy in which a single commodity is produced by a large number of competitive firms. The economy is inhabited by a finite number of heterogeneous and infinitely lived agents who supply the factor inputs labor and capital to the firms and use the resulting factor income to finance their consumption. Their objective is to maximize the discounted stream of utility derived from consumption subject to the constraint that their wealth is non-negative at each point in time. We derive a system of two difference equations that describes the dynamic evolution of capital stocks and show by means of a robust example that this system can have a snap-back repeller and, consequently, can generate chaotic dynamics.

Homoclinic bifurcations in n-dimensional endomorphisms, due to expanding periodic points

The study of bifurcation mechanisms which determine the transition from regular to chaotic behaviour (or “route to chaos”) in endomorphisms (maps with a nonunique inverse) has reached a good level of theoretical knowledge for one-dimensional maps. We refer to the book by Mira [l] as a review of the main results in this research area. The situation is quite different as regards endomorphisms of dimensions higher than one. A few results have been worked out until now [2-81, mainly in the simplest cases of two-dimensional endomorphisms. However, many mathematical models of interest in the applications are ultimately described by maps with a nonunique inverse. Clearly, in the plane IR2, or in higher-dimensional spaces, we can no longer take advantage of the Koenigs-Lemeray graphical construction, and of the qualitative behaviour of the graphs of powers of t(x) (t’(x), t3(x), . ..). which are powerful tools to identify the local and global bifurcations of one-dimensional endomorphisms, x’ = t(x), x E IR’. However, to study bifurcation mechanisms in IR”, n 2 2, we make use of properties which are the natural generalizations of those used in the one-dimensional case. Indeed, a basic tool for studying the dynamics and bifurcation mechanisms of endomorphisms is provided by (the analysis of) the critical manifolds, first introduced by Gumowski and Mira in 1965 (see [2] and references therein). The critical manifolds are the natural generalization to IR”, n 2 2, of the local extrema (also called critical points of rank-l) of one-dimensional endomorphisms. In this work, we deal with homoclinic bifurcations in endomorphisms of IR” (n 1 l), due to expanding periodic points. As in one-dimensional endomorphisms, we will see that critical manifolds play a pre-eminent role. Our purpose is to contribute to the understanding of the bifurcation mechanisms related to homoclinic orbits occurring in n-dimensional endomorphisms, mechanisms strictly related to the nonuniqueness of the inverse function, that is, typical of endomorphisms, not present in invertible maps, although similar to those occurring in these maps. In particular, this work concerns the global bifurcation (homoclinic bifurcations) occurring when a repelling fixed point becomes a “snap-back repeller”t (SBR henceforth), and in general when there exists a critical orbit homoclinic to an expanding

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouche to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.

CHUA'S CIRCUIT WITH A DISCONTINUOUS NONLINEARITY

A discrete map can be obtained from Chua's circuit1 with a discontinuous nonlinearity. Chaotic behavior is observed in the simulation of Chua's circuit with a discontinuous piecewise-linear element. The Poincaré map can be used to study the stability behavior and opens the possibility for analyzing the chaotic behavior by determining the existence of a snap-back repeller.

Snapback repellers and semiconjugacy for iterated entire mappings

A theorem of Poincare guarantees existence of the local conjugacy of an entire analytic mapping with an hyperbolically unstable fixed point to the linearized mapping. Since the local conjugacy can be extended to a global conjugacy, it is a valuable tool for the global study of dynamics. Especially we focus on snapback repellers which are defined as entire orbits which tend to an unstable fixed point in the past and snap back to the same fixed point. Snapback repellers correspond to the zeros of the semiconjugacy. It turns out that in general there exist infinitely many snapback points and for each one of them there exist infinitely many snapback repellers. The exceptional classes of functions with a different behavior are characterized. The proof exploits the Theorem of Picard about the range of values that an analytic function assumes near an essential singularity. Furtheron, we related the multiplicity of the zeros of the semiconjugacy to the occurrence of critical points in the corresponding snapback repeller. For quadratic mappings and their iterates, the zeros of the semiconjugacy have at most multiplicity two.

Chaos and topological entropy in dimension n>1

AbstractIn this paper we present a generalization to higher dimensions of the techniques for computation of the entropy of graphs in dimension one. Following these methods, we obtain a lower bound for the topological entropy of a differentiable map F:ℝn→ℝn possessing a snap-back repeller.

Chaotic behavior in piecewise-linear sampled-data control systems

This paper discusses the sufficient conditions for chaotic behavior in piecewise-linear sampled-data control systems by applying Shiraiwa-Kurata's theorem. First, it is shown that a discrete-time system with a piecewise-linear element is chaotic if the lower-dimensional system induced from the original system has a snapback repeller. Next, this result is applied to a piecewise-linear sampled-data control system. Finally, the chaotic region for a two-dimensional sampled-data control system with a dead zone element is studied, and two types of transition from a fixed point to a chaotic attractor are studied by numerical simulation.

The evolution of crisis: Bifurcation and demise of a snapback repeller

This paper studies the dynamics of members of the two-parameter family of maps x → μx(1 − x v ), emphasizing the evolution from snapback repeller to crisis bifurcations. The example of the square root map v = 1 2 is taken to represent the subfamily where v is fixed and taken from the range 1 2 ≤ v ≤ 1 . A map from such a subfamily is shown to be conjugate with a map with negative Schwarzian derivative. This allows a characterization of crisis as the demise of a snapback repeller on a proper subinterval.

Snap-back repellers and chaos in a discrete population model with delayed recruitment

Snap-back repellers and chaos in a discrete population model with delayed recruitment

Chaos in the cubic mapping

Motivated by a problem in genetics involving one locus with two alleles, R. M. May gave the example of the family of cubic maps x→ ax 3 + (1− a) x of the interval [-1,1] and established the existence of a bifurcating sequence of cycles of period 2 n over an interval of parameter values. This paper extends the analysis of May beyond that region. We find a critical value a ∗ beyond which the map has a snapback repeller and hence chaotic behavior; this value further marks the onset of the first cycle of odd order >1. When a=4 the map is onto the interval and we find an associated invariant measure.

A generalization of a theorem of marotto

In 1975, Li and Yorke [3] found the following fact. Let f: I→ I be a continuous map of the compact interval I of the real line R into itself. If f has a periodic point of minimal period three, then f exhibits chaotic behavior. The above result is generalized by F.R. Marotto [4] in 1978 for the multi-dimensional case as follows. Let f: Rn → Rn be a differentiate map of the n-dimensional Euclidean space Rn (n ≧ 1) into itself. If f has a snap-back repeller, then f exhibits chaotic behavior.In this paper, we give a generalization of the above theorem of Marotto. Our theorem can also be regarded as a generalization of the Smale’s results on the transversal homoclinic point of a diffeomorphism.