International Journal of Bifurcation and Chaos

It has been argued that Lyapunov exponents as a measure of predictability are of limited value because they only provide a global average. Characterizing an attractor by a distribution of times for initial uncertainties to increase by a factor of q has been suggested as a more useful alternative. These have found favor in some applications, despite assumptions of the fictitious perfect model scenario. Here, an electronic circuit, which offers a good test-bed for addressing predictability in the imperfect model scenario, is presented. A novel measure of predictability is presented and implications of model imperfection on characterizing the dynamics of chaotic systems are discussed. It is a case for a multiple model approach.

This paper provides a topological dynamics perspective on the full bifurcation unfolding in unimodal mappings. We present a bundle structure, visualized as a bundle plot, to show the evolution of symbolic space as we vary a system parameter. The bundle plot can be viewed as a limit process of an assignment plot, which are line assignments between points from two dynamical systems. Such line assignments are determined by a commuter, which is a coordinates transformation function that satisfies a commuting relationship but not necessarily a homeomorphism. The bundle structure is studied by understanding the implication of the system's qualitative changes. In addition, the case of the bundle plot with higher dimensional parameter variation is also considered. A main concern in the bundle plot is a special structure, called "joint", which determines a critical value of the parameter where the kneading sequence becomes periodic.

This paper reports the detailed parameter space study of the nonlinear dynamical behaviors and their control in a time-delay digital tanlock loop (TDTL). At first, we explore the nonlinear dynamics of the TDTL in parameter space and show that beyond a certain value of loop gain parameter the system manifests bifurcation and chaos. Next, we consider two variants of the delayed feedback control (DFC) technique, namely, the time-delayed feedback control (TDFC) technique, and its modified version, the extended time-delayed feedback control (ETDFC) technique. Stability analyses are carried out to find out the stable phase-locked zone of the system for both the controlled cases. We employ two-parameter bifurcation diagrams and the Lyapunov exponent spectrum to explore the dynamics of the system in the global parameter space. We establish that the control techniques can extend the stable phase-locked region of operation by controlling the occurrence of bifurcation and chaos. We also derive an estimate of the optimum parameter values for which the controlled system has the fastest convergence time even for a larger acquisition range. The present study provides a necessary detailed parameter space study that will enable one to design an improved TDTL system.

We introduce a family of area preserving generalized baker's transformations acting on the unit square and having sharp polynomial rates of mixing for Hölder data. The construction is geometric, relying on the graph of a single variable "cut function". Each baker's map B is nonuniformly hyperbolic and while the exact mixing rate depends on B, all polynomial rates can be attained. The analysis of mixing rates depends on building a suitable Young tower for an expanding factor. The mechanisms leading to a slow rate of correlation decay are especially transparent in our examples due to the simple geometry in the construction. For this reason, we propose this class of maps as an excellent testing ground for new techniques for the analysis of decay of correlations in non-uniformly hyperbolic systems. Finally, some of our examples can be seen to be extensions of certain 1D non-uniformly expanding maps that have appeared in the literature over the last twenty years, thereby providing a unified treatment of these interesting and well-studied examples.

The quantization behaviors of a sliding-mode control system with state feedback when the measurement of the state is quantized are studied. Some inherent dynamical properties for the quantized SMC systems are analyzed. The definition and the occurrence condition of "quantized sliding mode" are proposed. Theoretical results are illustrated with simulation examples.

We numerically investigate the role of the number of equilibrium points (N) on characteristic features of stochastic and coherence resonances in a modified Chua's circuit model equation capable of generating multi-scroll orbits. Both types of resonances are found for 1 < N ≤ 20. Almost periodic switching between the scroll orbits is observed at resonance. The values of the signal-to-noise ratio (used to quantify stochastic resonance), the autocorrelation function and the ratio of the dominant peak and its relative width in the power spectrum of the output signal (used to characterize coherence resonance) at resonance increase with N, reaching a maximum at N = 6 and then decrease. The mean residence time of a scroll orbit at resonance decreases rapidly with N. We show that both resonances in the multi-scroll orbits can be useful for weak signal detection.

In this paper, various notions of chaos in the Biebutov system are studied. First, the Biebutov system is generalized to the weighted Biebutov system. Meanwhile, we introduce the concept of continuous distributional chaos for the weighted Biebutov system and prove that it exhibits continuous distributional chaos and Li–Yorke sensitivity. Finally, we prove that the weighted Biebutov system is mixing and Devaney chaotic.

This paper is devoted to the synchronization of a dynamical system defined by two different coupling versions of two identical piecewise linear bimodal maps. We consider both local and global studies, using different tools as natural transversal Lyapunov exponent, Lyapunov functions, eigenvalues and eigenvectors and numerical simulations. We obtain theoretical results for the existence of synchronization on coupling parameter range. We characterize the synchronization manifold as an attractor and measure the synchronization speed. In one coupling version, we give a necessary and sufficient condition for the synchronization. We study the basins of synchronization and show that, depending upon the type of coupling, they can have very different shapes and are not necessarily constituted by the whole phase space; in some cases, they can be riddled.

The concept of memory element or memory system is generalized to any device or system with memory effect in this paper. With the general definition of memory element, a new memory element model is proposed and its properties are studied. Based on the memory element, a simple chaotic memory system is constructed and its complex dynamics is depicted. These results verified by circuit experimental observations and PSIM simulations indicate that the proposed memory element has a pinched hysteresis loop characteristics for a periodic input, which depends on the initial state of memory element, and the amplitude and frequency of the applied input; the proposed chaotic memory system presents complex dynamics depending on system parameters and the initial state of memory element.

The Muthuswamy–Chua system

This paper is concerned with the bifurcation of limit cycles of a class of quadratic reversible Lotka–Volterra system with b = -1/3. By using the Chebyshev criterion to study the number of zeros of Abelian integrals, we prove that this system has at most two limit cycles produced from the period annulus around the center under quadratic perturbations, which provide a positive answer for a case of the conjecture proposed by S. Gautier et al.

In the development of Drosophila wing disc, morphogen Dpp, which is a signaling molecule from a local region and disperses into anterior and posterior compartments, builds up a gradient with precise pattern information. Experiments have demonstrated that the key genes (brk, dad, omb and sal) and phosphorylated protein (pMad), which are activated by Dpp signaling molecules and form the gradients of the corresponding proteins of these genes, direct and control the spatial pattern of the wing disc. However, the regulatory network of these genes are in complex and nonlinear interaction with upstream regulators and downstream targets. In this paper, the mathematical model is built according to the regulatory relationships of these key genes. The stabilities of the gradients of these corresponding proteins are investigated. Furthermore, numerical simulations show that these gradients are robust with respect to some major reaction rates in this regulatory network.

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2) symmetry is perturbed by imperfection terms that break a continuous translation symmetry. The parameters of the fully symmetric normal form are fixed to values for which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is attracted by a chaotic attractor resulting from a period doubling cascade of periodic orbits. By using symmetry-adapted variables, the dimension of the phase space of the normal form is reduced to four and the dimension of the perturbed normal form is reduced to five. In the reduced phase space, periodic solutions are revealed as fixed points, and quasiperiodic solutions as periodic orbits. For the perturbed normal form, parameter regimes with different types of chaotic dynamics are identified when the imperfection parameter is varied. The characteristics of this complex dynamics are symmetry breaking and increasing, various period doubling cascades, intermittency and crises, and switching between symmetry-conjugated chaotic saddles. In particular, the perturbed system serves as a low dimensional model for the complicated switching dynamics found in simulations of the globally coupled system of Ginzburg–Landau equations extending the 𝕆(2) × 𝕆(2)-symmetric normal form to account for spatial modulations. In addition, this system can be considered as a low dimensional model for the dynamics of perturbed waves in anisotropic systems with imperfect geometries due to the presence of sidewalls.

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. In this article, we make a global study of the family of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.

In this paper, all possible sparse chemical reaction network structures of a classical three-dimensional Lorenz system are computed assuming a given chemical complex set. The original nonkinetic equations are transformed into kinetic form using two different approaches: firstly, using a state-dependent time-rescaling and secondly, by applying the theory of X-factorable systems. Using the notions of core reactions and core complexes, an effective optimization-based computation approach is proposed for the calculation of all structurally different sparse reaction graphs. The resulting structures are briefly analyzed and compared from a structural point of view.

In this work, we present a bifurcation analysis of a network of symmetrically coupled units modeling central pattern generators for quadruped locomotion. Here, we show a reduced model and characterize its dynamics and the dependence of the model behavior when one of the parameters is varied.

In this paper, we consider the planar system ẋ = -yF(x, y) + εP(x, y), ẏ = xF(x, y) + εQ(x, y), where the set {F(x, y) = 0} consists of m nonzero points (a_i, b_i)(i = 1, …, m) with multiple multiplicities, P(x, y) and Q(x, y) are arbitrary real polynomials. We study the number of limit cycles bifurcating from the periodic annulus surrounding the origin by using Abelian integrals and residue integration.

This paper proposes a Filippov epidemic model with piecewise continuous function to represent the enhanced vaccination strategy being triggered once the proportion of the susceptible individuals exceeds a threshold level. The sliding bifurcation and global dynamics for the proposed system are investigated. It is shown that as the threshold value varies, the proposed system can exhibit variable sliding mode domains and local sliding bifurcations including boundary node (focus) bifurcation, double tangency bifurcation and other sliding mode bifurcations. Model solutions ultimately approach either one of two endemic states for two structures or the pseudo-equilibrium on the switching surface, depending on the threshold level. The findings indicate that proper combinations of threshold level and enhanced vaccination rate based on threshold policy can lead disease prevalence to a previously chosen level if eradication of disease is impossible.

Sigmoidal functions are usually used to characterize the behavior of dynamical systems, in particular, for neural networks. Recently, multilevel piecewise linear functions have been employed in cellular neural networks (CNN). In this paper, we first use the inverse trigonometric function, tan^-1(x), to generate a series of trigonometric functions to obtain one-, two- and three-directional multi-scroll integer and fractional order chaotic attractors. Then, based upon the properties of the chaotic systems, simple feedback control laws are designed to stabilize or synchronize the integer and fractional order chaotic systems. Numerical simulations are presented to demonstrate the applicability of theoretical predictions.

In this paper, the chaos multiscale-synchronization between two different Fractional-order Hyperchaotic System (FOHCS)s have been investigated. The Lü-like and its FOHCS are also studied. The Lü-like FOHCS is controlled to be multiscale-synchronization with Liu FOHCS and new Lorenz FOHCS, respectively. The analytical conditions for the multiscale-synchronization of these pairs of different FOHCSs are derived by utilizing Laplace transform. Furthermore, multiscale-synchronization between two different FOHCSs is achieved by utilizing the different linear and nonlinear feedback control method in a short period and both remain in chaotic states. Numerical simulations are used to verify the theoretical analysis using different values of the fractional-order parameter and the different linear and nonlinear control functions.

The recurrence rate and determinism are two of the basic complexity measures studied in the recurrence quantification analysis. In this paper, the recurrence rate and determinism are expressed in terms of the correlation sum, and strong laws of large numbers are given for them.

We investigate errorless and perfectly reproducible coding of real sound data by means of one-dimensional cellular automata with two states and three neighbors related with the viewpoint of "rule dynamics". In our previous studies, we have shown that only two rules in 1-2-3 CA are sufficient to generate all the patterns consisting of 16 bits. Moreover, it was shown that the digital data of several spoken words and music can be reproduced perfectly. However, the reason why two-rules set of 1-2-3 CA can generate errorless coding for digital sound data has been unknown. In addition, the potentiality of the description method, data compressibility and description ability, has been unknown. In the present paper, therefore, we present the proof that the two-rules set can generate arbitrary bit-pattern sequences. From computer experiments, the two-rules set of (#90, #180) is practical in realizing compressive coding for digital spoken-words and music data without reproducing errors.