International Journal of Bifurcation and Chaos

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN^†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible.

The remaining 86 CA rules are time-irreversible in the sense that N and N^† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time.

A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N^†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology.

Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors.

Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,.

or its inverse map.

These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes.

Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

This paper presents a novel result on the effect of coupling through both analytical and numerical investigations on linearly coupled systems including chaotic and nonchaotic systems. It is found that when a single system has potential of oscillation but is currently in a "marginal" state to produce a limit cycle via Hopf bifurcation due to the change of a parameter, an appropriate coupling strength can excite the potential limit cycle such that the coupled system oscillates synchronously. Similarly, when a stable limit cycle is at the "margin" of a chaotic attractor in a single system, a certain coupling strength can induce the potential chaotic attractor such that the coupled system has a synchronous chaotic behavior. This excitation mechanism is different from the traditional function of coupling in that the latter mainly drives the coupled system to synchronize with the ongoing dynamics of a single system but does not recover its disappearing dynamics. This newly observed synchronization is called coherent synchronization to distinguish it from various common types of synchronization. Several numerical examples are presented for quantitative description of this interesting phenomenon.

Recently, an effective method for realizing linearly separable Boolean functions via Cellular Neural Networks (CNN), called the threshold bifurcation method, was introduced, with a CNN gene bank of four variables established [Chen & Chen, 2005]. Based on this success, the present paper is to further explore the realization of all linearly separable Boolean functions of five variables via CNN with von Neumann neighborhoods. This paper provides: (i) important and essential relations among the genes (or templates) and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set, (ii) a neat truth table of uncoupled CNN with five input variables, (iii) 94572 linearly separable Boolean functions (LSBF) in the family of 2²⁵ = 4.294967296 × 10⁹ Boolean functions of five variables, realizable by a single CNN, and (iv) all 94572 CNN linearly separable Boolean genes (LSBG), which can be determined to form the CNN gene bank of five variables.

In this paper we study the existence, uniqueness and stability of periodic solutions for a two-neuron network system with or without external inputs. The system consists of two identical neurons, each possessing nonlinear feedback and connected to the other neuron via a nonlinear sigmoidal activation function. In the absence of external inputs but with appropriate conditions on the feedback and connection strengths, we prove the existence, uniqueness and stability of periodic solutions by using the Poincaré–Bendixson theorem together with Dulac's criterion. On the other hand, for the system with periodic external inputs, combining the techniques of the Liapunov function with the contraction mapping theorem, we propose some sufficient conditions for establishing the existence, uniqueness and exponential stability of the periodic solutions. Some numerical results are also provided to demonstrate the theoretical analysis.

An automatic procedure for generating colored two-dimensional symmetrical images based on the chemical reactions discrete chaotic dynamics (CRDCD) is proposed. The inverse problem of derivation of symmetrical images from CRDCD mathematical models was formulated and solved using a special type of genetic algorithm. Different symmetrical images corresponding to the solutions of a CRDCD mathematical model for which the parameters were obtained automatically by the proposed method are presented.

This paper investigates the synchronization of chaotic systems subject to parameter uncertainties. Based on the fuzzy-model-based approach, a switching controller will be proposed to deal with the synchronization problem. The stability conditions will be derived based on the Lyapunov approach. The tracking performance and parameter design of the proposed switching controller will be formulated as a generalized eigenvalue minimization problem which can be solved numerically using some convex programming techniques. Simulation examples will be given to show the effectiveness of the proposed approach.

In this paper, we present two new schemes to synchronize a driven chaotic system with the driving chaotic system in the presence of parametric uncertainties and parametric mismatch between the two systems. A controller is designed for the driven system based on the bounds of the uncertainties. Sufficient conditions are derived to ensure synchronization of the two systems. Numerical results also show the robustness and effectiveness of the proposed schemes.

A method for estimating the parameters of continuous-time Lur'e systems is proposed. The method is based on the derivation of an approximate, discrete-time model, in the form of a scalar difference equation, which can directly be used for data fitting. The discretization can be obtained through any (fixed-step) integration scheme: in the paper, the performance of a simple scheme (the trapezoidal rule) and a more complex one (the Runge–Kutta rule) are compared. The method, tested on three different examples (Duffing system, an electrical circuit and Chua's circuit), proves to be capable of obtaining a rapid parameter estimation with an acceptable level of accuracy.

In this paper, we prove that chaos in the sense of Li–Yorke and of Devaney is prevalent in discrete systems admitting the so-called heteroclinical repellers, which are similar to the transversely heteroclinical orbits in both continuous and discrete systems and are corresponding to the snap-back repeller proposed by Marotto for proving the existence of chaos in higher-dimensional systems. In addition, the concept of heteroclinical repellers is generalized to be applicable to the case with degenerate transformations. In the end, some illustrative examples are provided to illustrate the theoretical results.

This paper presents an iterative method for locating invariant tori of maps. Specifically, through the introduction of a shift function on the torus corresponding to the projection of the dynamics on the torus onto toral parameters, a discrete system of equations may be formulated whose solution approximates the spatial location of the torus. As invariant tori of continuous flows may be considered invariant under the application of the flow for a fixed time, or may correspond to invariant tori (of one less dimension) of suitably introduced Poincaré maps, the methodology applies to discrete and continuous dynamical systems alike. Moreover, the insensitivity of the method to the local stability characteristics of the invariant torus as well as to the precise nature of the flow on the torus imply, for example, that the method can be employed for the continuation of unstable invariant tori on which the dynamics are attracted to a periodic trajectory as long as the torus is sufficiently smooth. The proposed methodology as well as reduced formulations that result from a priori knowledge about the invariant torus are illustrated through some sample dynamical systems.

We define a locally Sierpinski Julia set to be a Julia set of an elliptic function which is a Sierpinski curve in each fundamental domain for the lattice. In order to construct examples, we give sufficient conditions on a lattice for which the corresponding Weierstrass elliptic ℘ function is locally connected and quadratic-like, and we use these results to prove the existence of locally Sierpinski Julia sets for certain elliptic functions. We give examples satisfying these conditions. We show this results in naturally occurring Sierpinski curves in the plane, sphere and torus as well.

This Letter contributes to the topic of chaos computing by introducing a new chaos-based implementation of sequential gates. In particular, the early results reported herein show that a chaos-based SR flip–flop can be obtained from two cross-coupled NOR gates implemented by a single Chua's circuit.

This study presents an experimental investigation of impulsive synchronization between two Mixed-mode chaotic circuits. The proposed impulsive synchronization scheme has been tested by applying a switching process with frequency range from 500 Hz to 1 MHz. Experimental results recorded with a digital storage oscilloscope revealed that the proposed scheme can be impulsively synchronized with good quality for switching frequencies higher than 10 kHz.

In this paper, we report on pattern formation occurring on the ACE16k CNN chip. The CNN chip can be programmed with a cloning template in order to generate spiral waves and autowaves. The waves diffract from internal sources which cannot be relocated on the network. However, by using initial and/or input images, sources (external sources) can be located at any place on the network. Furthermore a competition between autowaves generated by external and internal sources is observed. Propagation of autowaves on the inhomogeneous CNN array, formed by the fixed-state map, is presented.

Dynamics of spiral waves in perturbed, e.g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spirals Response Functions, which are eigenfunctions of the adjoint linearized problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh–Nagumo model, and show that they are effectively localized in the vicinity of the spiral core.

As a powerful cryptanalysis tool, the method of return-map attacks can be used to extract secret messages masked by chaos in secure communication schemes. Recently, a simple defensive mechanism was presented to enhance the security of chaotic parameter modulation schemes against return-map attacks. Two techniques are combined in the proposed defensive mechanism: Multistep parameter modulation and alternative driving of two different transmitter variables. This paper restudies the security of this proposed defensive mechanism against return-map attacks, and points out that the security was much over-estimated in the original publication for both ciphertext-only attack and known/chosen-plaintext attacks. It is found that a deterministic relationship exists between the shape of the return map and the modulated parameter, and that such a relationship can be used to dramatically enhance return-map attacks thereby making them quite easy to break the defensive mechanism.

Synchronization patterns in chains of N bi-directionally delayed coupled systems with delayed feedback are studied. Each system is hyperchaotic when decoupled from the chain. It is shown that chains with odd or even number of cites N display different spatial patterns of stable exact synchronization. When N is odd the only stable pattern of exact synchronization is between all of the units. When N is even, next to the nearest neighbors could become exactly synchronized, with the dynamics of the nearest neighbors related in a more complicated way. Sufficiently strong coupling leads to the nearest neighbor synchronization also for even N. No other patterns have been observed.

Weak periodic perturbation has long been used to suppress chaos in dynamical systems. In this paper, however, we demonstrate that weak periodic or quasiperiodic perturbation can also be used to induce chaos in nonchaotic parameter ranges of chaotic maps, or to enhance the already existing chaotic state. Two kinds of chaotic maps, the period doubling system and the Hopf bifurcation system, are employed as basic models to analyze and compare in detail the different mechanisms of inducing and enhancing chaos in them. In addition, a special kind of intermittency characterized by its periodicity is found for the first time in periodically perturbed Henon map, and reasonable speculations are presented to explain its complicated dynamics.

In this note we analyze the periodicity of an implicit nonlinear difference equation subject to perturbations and discontinuity. We prove the existence of multiple periodic solutions under certain conditions, and derive the periods which are functions of the sampling period and the period of the perturbation. In this work, we further present a simulation algorithm which enables the computation of the solution for the implicit difference equation in an iterative manner. The Van der Pol circuit is used as an illustrative example.