International Journal of Bifurcation and Chaos

Chaos and intermittency are studied for the system of globally coupled, complex Ginzburg–Landau equations governing the dynamics of extended, two-dimensional anisotropic systems near an oscillatory (Hopf) instability of a basic state with two pairs of counterpropagating, oblique traveling waves. Parameters are chosen such that the underlying normal form, which governs the dynamics of the spatially constant modes, has two symmetry-conjugated chaotic attractors. Two main states residing in nested invariant subspaces are identified, a state referred to as Spatial Intermittency ($SI$) and a state referred to as Spatial Persistence ($SP$). The $SI$-state consists of laminar phases where the dynamics is close to a normal form attractor, without spatial variation, and switching phases with spatiotemporal bursts during which the system switches from one normal form attractor to the conjugated normal form attractor. The $SP$-state also consists of two symmetry-conjugated states, with complex spatiotemporal dynamics, that reside in higher dimensional invariant subspaces whose intersection forms the 8D space of the spatially constant modes. We characterize the repeated appearance of these states as (generalized) in–out intermittency. The statistics of the lengths of the laminar phases is studied using an appropriate Poincaré map. Since the Ginzburg–Landau system studied in this paper can be derived from the governing equations for electroconvection in nematic liquid crystals, the occurrence of in–out intermittency may be of interest in understanding spatiotemporally complex dynamics in nematic electroconvection.

We study spiral waves in a mathematical model of a nonlinear optical system with a feedback loop. Starting from a delayed scalar diffusion equation in a thin annulus with oblique derivative boundary conditions, we shrink the annulus and derive the limiting equation on a circle. Based on the explicitly constructed normal form of the Hopf bifurcation for the one-dimensional delayed scalar diffusion equation, we make predictions about the existence and stability of two-dimensional spirals that we verify in direct numerical simulations, observing pulsating and rotating spiral waves.

A model of quasiperiodic-chaotic neural networks is proposed on the basis of chaotic neural networks. A quasiperiodic-chaotic neuron exhibits quasiperiodic dynamics that an original chaotic neuron does not have. Quasiperiodic and chaotic solutions are exclusively isolated in the parameter space. The chaotic domain can be identified by the presence of a folding structure of an invariant closed curve. Using the property that the influence of perturbation is conserved in the quasiperiodic solution, we demonstrate short-term visual memory in which real numbers are acceptable for representing colors. The quasiperiodic solution is sensitive to dynamical noise when images are restored. However, the quasiperiodic synchronization among neurons can reduce the influence of noise. Short-term analog memory using quasiperiodicity is important in that it can directly store analog quantities. The quasiperiodic-chaotic neural networks are shown to work as large-scale analog storage arrays. This type of analog memory has potential applications to analog computation such as deep learning.

This paper studies two two-component shallow water wave models. From the dynamical systems approach and using the singular traveling wave theory developed by Li and Chen [2007], all possible bounded solutions (solitary wave solutions, pseudo-peakons, periodic peakons, as well as smooth periodic wave solutions) are obtained under different parameter conditions. More than six explicit exact parametric representations are derived. More interestingly, it was found that, for the two-component Camassa–Holm equations with constant vorticity, its $u$-traveling wave system has a pseudo-peakon wave solution. In addition, its $h$-traveling wave system has four families of uncountably infinitely many solitary wave solutions. The new results complete a recent study of Dutykh and Ionescu-Kruse [2019].

The reaction time delay in the transcription process depends on the concentration of the protein because the transportation of mRNA from the nucleus to the cytoplasm becomes saturated. Thus the gene regulatory network is a state-dependent delayed model. This study aims to provide some mathematical explanations for the dynamics of the system, such as the linear stability and periodic oscillation, using mathematical techniques, such as formal linearization, linear stability analysis, the method of multiple scale (MMS), and the normal form. First, Hopf bifurcation of the state-dependent delayed gene regulatory networks model in the gene expression is analyzed by the method of multiple scales (MMS). Mechanism of periodic oscillations is obtained by Hopf bifurcation. The findings show that when degradation effects of the mRNA and protein are very strong, the oscillatory gene expression disappears. Then, a more realistic version of the aforementioned model with both constant and state-dependent time delays is established due to the existence of the constant time delay in the protein degradation process. Its nonresonant double Hopf bifurcation is found and analyzed using MMS. Interesting complex dynamic phenomena, such as periodic, quasi-periodic, and global period-$2$ solutions, are also discovered. These observations indicate that both state-dependent delay and constant delay could induce richer dynamics of the system, and the modified model may potentially describe the real dynamical mechanism (both the transcription process and the degradation process) more accurately in the gene expression. The findings may provide important guidance or hints to understand the real dynamic mechanism of the gene expression process.

In this paper, a new spatiotemporal chaotic Parameter Uncertainty Mixed Coupled Map Lattice (PUMCML) is proposed. The Cantor diagonal matrix is generated from the Cantor set, and the ordered rotation scrambling strategy for this matrix is used to generate the scrambled image. Cantor set is a fractal system, so the Cantor set has a good effect on chaotic image encryption. The dynamic behavior of the PUMCML system is analyzed. The system has good chaotic property, so it is very suitable for chaotic image encryption. Using the PUMCML system, a diffusion strategy based on the mixture of Arnold and Logistic is proposed. Compared with other algorithms, the encryption effect of the proposed method is better and more secure.

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results.

In this paper, we study the unilateral global bifurcation from infinity in nonlinearizable eigenvalue problems for the one-dimensional Dirac equation. We show the existence of two families of unbounded continua of the set of nontrivial solutions emanating from asymptotically bifurcation intervals and having the usual nodal properties near these intervals.

In this paper, a couple map lattice (CML) model is used to study the spatiotemporal dynamics and Turing patterns for a space-time discrete generalized toxic-phytoplankton-zooplankton system with self-diffusion and cross-diffusion. First, the existence and stability conditions for fixed points are obtained by using linear stability analysis. Second, the conditions for the occurrence of flip bifurcation, Neimark–Sacker bifurcation and Turing bifurcation are obtained by using the center manifold reduction theorem and bifurcation theory. The results show that there exist two nonlinear mechanisms, flip-Turing instability and Neimark–Sacker–Turing instability. Moreover, some numerical simulations are used to illustrate the theoretical results. Interestingly, rich dynamical behaviors, such as periodic points, periodic or quasi-periodic orbits, chaos and interesting patterns (plaques, curls, spirals, circles and other intermediate patterns) are found. The results obtained in the CML model contribute to comprehending the complex pattern formation of spatially extended discrete generalized toxic-phytoplankton-zooplankton system.

Memristor can greatly enhance the complexity of a chaotic system because of its nonlinear characteristics. In this paper, three different memristor models are introduced to the Yang system. The chaotic attractors with single scroll and double scrolls can be obtained by adjusting the action intensities of three memristors and all the attractors inherit the scaling property of attractors of the Yang system. By employing the complex polynomials transformation method in the chaotic system to expand the number of scrolls of the system, the ring-shaped multiscroll attractors are generated, and the number of scrolls can be changed by adjusting the powers of complex polynomials, which show that the memristive system has flexible scalability. Next, a synchronization method for the multiscroll chaotic system is proposed. The generalized synchronization controller and parameter adaptive law are designed by employing sliding mode control. The sufficient conditions for synchronization are given by Lyapunov stability theory. This method can be applied to the synchronization of multiscroll systems generated by means of changing the state variables of the original system by function transformation and then adding the transformation matrix to the system. Compared with the existing synchronization method, this method has a wider scope of application, and it can synchronize two multiscroll chaotic systems with greater difference. In addition, the conditions to be satisfied in this method are simpler. Finally, the method proposed above is applied to the synchronization between a chaotic system with a ring-shaped eight-scroll attractor and a grid-shaped $4\times 4$-scroll attractor chaotic system with interference signals. The numerical simulation results verify the effectiveness of the method.

The Collatz–Wielandt formula obtained by Lothar Collatz (1942) and Helmut Wielandt (1950) provides a simple variational characterization of the Perron–Frobenius eigenvalue of certain types of matrices. We introduce a nonlinear generalization of the Collatz–Wielandt formula and show that it enables finding the so-called maximal saddle-node bifurcations of systems of nonlinear equations. As a consequence, we obtain an easily verifiable criterion for the detection of such saddle-node bifurcations. We illustrate our approach by examples of finite-difference approximations of nonlinear partial differential equations and a system of power flow.

A recent paper investigates the bifurcation diagrams involved with torus doubling and asserts that the chaotic attractors observed after torus doubling have two Lyapunov exponents that are exactly zero. Against this assertion, we claim that the absolute value of one of the calculated zero Lyapunov exponents is not exactly zero but is instead slightly positive, because successive torus doubling is constrained by a very small underlying parameter. We justify our position by calculating Lyapunov spectra precisely using an autonomous piecewise-linear dynamical circuit. Our numerical results show that one of the Lyapunov exponents is close to, but not exactly, zero. In addition, we consider coupled logistic and sine-circle maps whose dynamics express the fundamental mechanism that causes torus doubling, and we confirm that torus doubling occurs fewer times when the coupling parameter of this discrete dynamical system is relatively larger. Consequently, the absolute value of the second Lyapunov exponent of this discrete system does not approach zero after the accumulation of torus doubling when the coupling parameter is set to larger values.

We present a new solving procedure in this paper for Kelvin–Kirchhoff equations, considering the dynamics of a falling rigid rotating torus in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the rotating body, $I_1=I_2$.

The fundamental law of angular momentum conservation is used for the aforementioned solving procedure. The system of Euler equations for the dynamics of torus rotation is explored for an analytic way of presentation of the approximated solution (where we consider the case of laminar flow at slow regime of torus rotation). The second finding is that the Stokes boundary layer phenomenon on the boundaries of the torus also assumed additionally at the formulation of basic Kelvin–Kirchhoff equations (for which the analytical expressions for the components of fluid’s torque vector $\{T_2,T_3\}$ were obtained earlier). The results for calculating the components of angular velocity $\{{\mathrm{\Omega }}_i\}$ should then be used for full solving the momentum equation of Kelvin–Kirchhoff system. The trajectories of motion can be divided into, preferably, three classes: zigzagging, helical spiral motion, and the chaotic regime of oscillations.

In this paper, we propose a guideline for plotting the bifurcation diagrams of chaotic systems. We discuss numerical and mathematical facts in order to obtain more accurate and more elegant bifurcation diagrams. The importance of transient time and the phenomena of critical slowing down are investigated. Some critical issues related to multistability are discussed. Finally, a solution for fast obtaining an accurate sketch of the bifurcation diagram is presented. The solution is based on running the system for only one sample in each parameter value and using the system’s state in the previous value of the parameter as the initial condition.

In this paper, we investigate a three-parameter four-dimensional dynamical system, which is modeled by a set of four first-order nonlinear ordinary differential equations, each of which contains a crossed cubic term. Dynamical behaviors are characterized in the parameter space of the model. In fact, we use some cross-sections of a three-dimensional parameter-space, namely three related parameter planes, to locate regular and chaotic regions, as well as multistability regions. Lyapunov exponents spectra, bifurcation diagrams, and phase-space portraits are used to complete the analysis.

This article presents a unified four-dimensional autonomous chaotic system with various coexisting attractors. The dynamic behaviors of the system are determined by its special nonlinearities with multiple zeros. Two cases of nonlinearities with sine function of the system are discussed. The symmetrical coexisting attractors, asymmetrical coexisting attractors and infinitely many coexisting attractors in the system are numerically demonstrated. This shows that such a system has an ability to produce abundant coexisting attractors, depending on the number of equilibrium points determined by nonlinearities.

We present counterexamples illustrating that the characterization of the reversibility of linear cellular automata on finite triangular grids given by Uguz et al. [2017] and Uguz et al. [2019] is not valid, neither in the case of null boundary conditions nor in the case of periodic boundary conditions.

In recent years, research on synchronization between coupled chaotic circuits has attracted interest in a wide range of fields. This is because the synchronization of coupled chaotic circuits is a multidisciplinary phenomenon that occurs in various applications, such as broadband communication systems or secure communication.

In this study, we propose a coupled chaotic circuit network model with stochastic couplings. We investigate the synchronization phenomena observed for the proposed network using different network structures such as fully-coupled, random, small world and scale-free networks. We find that the same synchronization characteristics can be obtained for these networks with a dynamic topology as when the coupling strength is changed in static networks.

In this paper, we investigate the chaotic behaviors of the Chen system with Caputo–Hadamard derivative. First, we construct some practical numerical schemes for the Chen system with Caputo–Hadamard derivative. Then, by means of the variational equation, we estimate the bounds of the Lyapunov exponents for the considered system. Furthermore, we analyze the dynamical evolution of the Chen system with Caputo–Hadamard derivative based on the Lyapunov exponents, and we found that chaos does exist in the considered system. Some phase diagrams and Lyapunov exponent spectra are displayed to verify our analysis.

We introduce the concepts of sensitivity, multisensitivity, cofinite sensitivity, and syndetic sensitivity for nonautonomous dynamical systems on uniform spaces and obtain some sufficient conditions under which topological transitivity and dense periodic points imply sensitivity for nonautonomous systems on Hausdorff uniform spaces. We also study sensitivity and other stronger versions of sensitivity for the systems induced on hyperspaces and for the product of nonautonomous dynamical systems on uniform spaces.