International Journal of Bifurcation and Chaos

This 12th part of our Nonlinear Dynamics Perspective of Cellular Automata concludes a series of three articles devoted to CA local rules having robust periodic ω-limit orbits. Here, we consider only the two rules, and , constituting the third of the six groups in which we classified the 1D binary Cellular Automata. Among the numerous theoretical results contained in this article, we emphasize the complete characterization of the ω-limit orbits, both robust and nonrobust, of these two rules and the proof that period-3 and period-6 ω-limit orbits are dense for and , respectively. Furthermore, we will also introduce the fundamental concepts of perfect period-T orbitsets and riddled basins, and see how they emerge in rule .

As stated in the title, we also focus on permutive rules, which have been introduced in a previous installment of our series but never thoroughly studied. Indeed, we will review some of the well-known properties of such rules, like the surjectivity, examining their implications for finite and bi-infinite Cellular Automata.

Finally, we propose a new list of the 88 globally-independent local rules, which is slightly different from the one we have used so far but has the great advantage of being selected via a rigorous methodology and not an arbitrary choice. For the sake of completeness, we display in the appendix the basin tree diagrams and the portraits of the ω-limit orbits of the rules from this refined table which have not yet been reported in our previous articles.

We construct a statistical ensemble of games, wherein each independent subensemble we have two players playing the same game. We derive the mean payoffs per move of the representative players of the game, and we evaluate all the deterministic policies with finite memory. In particular, we show that if one of the players has a generalized tit-for-tat policy, the mean payoff per move of both players is the same, forcing the equalization of the mean payoffs per move. In the case of symmetric, noncooperative and dilemmatic games, we show that generalized tit-for-tat or imitation policies together with the condition of not being the first to defect, leads to the highest mean payoffs per move for the players. Within this approach, it can be decided which policies perform better than others. In particular, it shows that reciprocity in noncooperative iterated games forces equality of mean payoffs. We prove a simple ergodic theorem for symmetric and noncooperative games. The Prisoner's Dilemma and the Hawk–Dove games have been analyzed, and the equilibrium states of the infinitely iterated games have been determined. In infinitely iterated games with the player choosing their moves with equal probabilities, strict Nash solutions are not necessarily reachable equilibrium solutions of games.

We present a method for constructing dynamical systems with robust, stable limit cycles in arbitrary dimensions. Our approach is based on a correspondence between dynamics in a class of differential equations and directed graphs on the n-dimensional hypercube (n-cube). When the directed graph contains a certain type of cycle, called a cyclic attractor, then a stable limit cycle solution of the differential equations exists. A novel method for constructing regulatory systems that we call minimal regulatory networks from directed graphs facilitates investigation of limit cycles in arbitrarily high dimensions.

We identify two families of cyclic attractors that are present for all dimensions n ≥ 3: cyclic negative feedback and sequential disinhibition. For each, we obtain explicit representations for the differential equations in arbitrary dimension. We also provide a complete listing of minimal regulatory networks, a representative differential equation, and a bifurcation analysis for each cyclic attractor in dimensions 3–5. This work joins discrete concepts of symmetry and classification with analysis of differential equations useful for understanding dynamics in complex biological control networks.

The Repressilator is a genetic regulatory network used to model oscillatory behavior of more complex regulatory networks like the circadian clock. We prove that the Repressilator equations undergo a supercritical Hopf bifurcation as the maximal rate of protein synthesis increases, and find a large range of parameters for which there is a cycle.

We propose a new approach to investigate the frequency synchronization in networks of coupled oscillators. By making use of the theory of monotone dynamical systems, we show that frequency synchronization occurs in networks of coupled oscillators, provided the coupling scheme is symmetric, connected, and strongly cooperative. Our criterion is independent of the system size, the coupling strength and the details of the connections, and applies also to nonlinear coupling schemes.

In this paper, we consider bifurcation of limit cycles in near-Hamiltonian systems. A new method is developed to study the analytical property of the Melnikov function near the origin for such systems. Based on the new method, a computationally efficient algorithm is established to systematically compute the coefficients of Melnikov function. Moreover, we consider the case that the Hamiltonian function of the system depends on parameters, in addition to the coefficients involved in perturbations, which generates more limit cycles in the neighborhood of the origin. The results are applied to a quadratic system with cubic perturbations to show that the system can have five limit cycles in the vicinity of the origin.

We have identified several prominent routes, namely, fractalization, fractalization followed by intermittency, intermittency and Heagy–Hammel routes, for the birth of strange nonchaotic attractors (SNAs) in a quasiperiodically forced electronic system with nonsinusoidal (square wave) force as one of the quasiperiodic forces [Senthilkumar et al., 2008]. In addition, a new bubbling route has also been identified in this circuit. Although some of these prominent routes have been reported experimentally [Thamilmaran et al., 2006] in a quasiperiodically forced electronic circuit with both the forcings as sinusoidal forces, experimental identification of all these routes is reported here in a quasiperiodically forced electronic circuit with one of the forcings as a nonsinusoidal (square wave) force. The birth of SNAs by these routes are characterized from both the experimental and numerical data by the maximal Lyapunov exponents and their variance, Poincaré maps, Fourier amplitude spectra, spectral distribution functions and the distribution of finite-time Lyapunov exponents.

We show that each weakly mixing group action on a dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, we prove that no nilpotent group action on a dendrite is weakly mixing. At last, we show that each dendrite admits no chaotic group actions.

In this paper, we discuss an efficient iterative method for the estimation of the chief dynamical invariants of chaotic systems based on stochastically stable piecewise affine maps (e.g. the invariant measure, the Lyapunov exponent as well as the Kolmogorov–Sinai entropy). The proposed method represents an alternative to the Monte-Carlo methods and to other methods based on the discretization of the Frobenius–Perron operator, such as the well known Ulam's method. The proposed estimation method converges not slower than exponentially and it requires a computation complexity that grows linearly with the iterations. Referring to the theory developed by C. Liverani, we discuss a theoretical tool for calculating a conservative estimation of the convergence rate of the proposed method. The proposed approach can be used to efficiently estimate any order statistics of a symbolic source based on a piecewise affine mixing map.

We study some of the most commonly used mutual information estimators, based on histograms of fixed or adaptive bin size, k-nearest neighbors and kernels and focus on optimal selection of their free parameters. We examine the consistency of the estimators (convergence to a stable value with the increase of time series length) and the degree of deviation among the estimators. The optimization of parameters is assessed by quantifying the deviation of the estimated mutual information from its true or asymptotic value as a function of the free parameter. Moreover, some commonly used criteria for parameter selection are evaluated for each estimator. The comparative study is based on Monte Carlo simulations on time series from several linear and nonlinear systems of different lengths and noise levels. The results show that the k-nearest neighbor is the most stable and less affected by the method-specific parameter. A data adaptive criterion for optimal binning is suggested for linear systems but it is found to be rather conservative for nonlinear systems. It turns out that the binning and kernel estimators give the least deviation in identifying the lag of the first minimum of mutual information from nonlinear systems, and are stable in the presence of noise.

This work shows the experimental implementation of a chaotic communication system based on two Chua's oscillators which are synchronized by Hamiltonian forms and observer approach. The chaotic communication scheme is realized by using the commercially available positive-type second generation current conveyor (CCII+), which is included into the AD844 device. As a result, experimental measurements are provided to demonstrate the suitability of the CCII+ to implement chaotic communication systems.

This paper proposes the construction of a new chaotic pseudorandom bit generator, which forms the main building block of a chaotic stream cipher. The design of the algorithm is based on a single chaotic map whose numerical orbit indirectly contributes towards the generation of the keystream. The latter is produced from the numerical orbit by applying a technique that searches for iterates in specific intervals [a,b], for some real numbers a and b, and outputs 0 or 1 based on the iterate preceding the targeted iterate. The generator suggested here is built up from a quadratic map. We analyze the cycle length of the keystreams and investigate the resistance of the generator to well-known cryptanalytic attacks. Furthermore, the statistic characteristics of the keystreams are examined numerically using the NIST statistical test suite. The numerical and theoretical results demonstrate that the proposed technique results in generating keystreams possessing very good cryptographic properties and high level of security against existing cryptanalytic attacks. Empirical results show that the search technique leads to the generation of keystreams possessing good randomness properties when applied to any chaotic map whose orbits have good randomness properties such as the quadratic map, tent map and sawtooth map.

It has been observed that Internet gateways employing Transport Control Protocol (TCP) and the Random Early Detection (RED) control algorithm may exhibit instability and oscillatory behavior. Most control methods proposed in the past have been based on analytical models that rely on statistical measurements of network parameters. In this paper, we apply the detrended fluctuation analysis (DFA) method to analyze stability of the TCP-RED system. The DFA is used to analyze time-series data and generate power-law scaling exponents, which indicate the long-range correlations of the time series. We quantify the stability of the TCP-RED system by examining the variation of the DFA power-law scaling exponent when the system parameters are varied. We also study the long-range power-law correlations of TCP window periods.

A nonadditive generalization of Klimontovich's S-theorem [Bagci, 2008] has recently been obtained by employing Tsallis entropy. This general version allows one to study physical systems whose stationary distributions are of the inverse power law in contrast to the original S-theorem, which only allows exponential stationary distributions. The nonadditive S-theorem has been applied to the modified Van der Pol oscillator with inverse power law stationary distribution. By using nonadditive S-theorem, it is shown that the entropy decreases as the system is driven out of equilibrium, indicating self-organization in the system. The allowed values of the nonadditivity index q are found to be confined to the regime (0.5,1].

A functional version of the LaSalle invariance principle is introduced. Rather than the usual pointwise Lyapunov-like functions, this extended version of the principle uses specially constructed functionals along system trajectories. This modification enables the original principle to handle not only autonomous, but also some nonautonomous systems. The new theoretical result is used to study robust synchronization of general Liénard-type nonlinear systems. The new technique is finally applied to coupled chaotic van der Pol oscillators to achieve synchronization. Numerical simulation is included to demonstrate the effectiveness of the proposed methodology.

A system of many coupled oscillators on a network can show multicluster synchronization. We obtain existence conditions and stability bounds for such a multicluster synchronization. When the oscillators are identical, we obtain the interesting result that network structure alone can cause multicluster synchronization to emerge even when all the other parameters are the same. We also study occurrence of multicluster synchronization when two different types of oscillators are coupled.