International Journal of Bifurcation and Chaos

In this paper, we will consider a family of 2-periodic nonautonomous dynamical systems, generated by the alternate iteration of two stunted tent maps and study its bifurcation skeleton. We will describe the bifurcation phenomena along and around the bones accomplished with the combinatorial data furnished by the respective symbolic dynamics.

We report using Clarke’s concept of generalized differential and a modification of Floquet theory to nonsmooth oscillations, the occurrence of discontinuity induced Hopf bifurcations and Neimark–Sacker bifurcations leading to quasiperiodic attractors in a memristive Murali–Lakshmanan–Chua (memristive MLC) circuit. The above bifurcations arise because of the fact that a memristive MLC circuit is basically a nonsmooth system by virtue of having a memristive element as its nonlinearity. The switching and modulating properties of the memristor which we have considered endow the circuit with two discontinuity boundaries and multiple equilibrium points as well. As the Jacobian matrices about these equilibrium points are noninvertible, they are nonhyperbolic, some of these admit local bifurcations as well. Consequently when these equilibrium points are perturbed, they lose their stability giving rise to quasiperiodic orbits. The numerical simulations carried out by incorporating proper discontinuity mappings (DMs), such as the Poincaré discontinuity map (PDM) and zero time discontinuity map (ZDM), are found to agree well with experimental observations.

In this paper, we provide for the first time rigorous mathematical results regarding the rich dynamics of piecewise linear memristor oscillators. In particular, for each nonlinear oscillator given in [Itoh & Chua, 2008], we show the existence of an infinite family of invariant manifolds and that the dynamics on such manifolds can be modeled without resorting to discontinuous models. Our approach provides topologically equivalent continuous models with one dimension less but with one extra parameter associated to the initial conditions. It is possible to justify the periodic behavior exhibited by three-dimensional memristor oscillators, by taking advantage of known results for planar continuous piecewise linear systems.

The analysis developed not only confirms the numerical results contained in previous works [Messias et al., 2010; Scarabello & Messias, 2014] but also goes much further by showing the existence of closed surfaces in the state space which are foliated by periodic orbits. The important role of initial conditions that justify the infinite number of periodic orbits exhibited by these models, is stressed. The possibility of unsuspected bistable regimes under specific configurations of parameters is also emphasized.

In this paper, we first present a survey of the known results on limit cycles and center conditions for Liénard differential systems. Next we propose a generalization of such systems and we study their center conditions and the number of small-amplitude limit cycles that can bifurcate from the origin. Computing the focal values and using Gröbner bases we find the center conditions for such systems up to a certain degree. We also establish a conjecture about the center conditions for such systems when they have arbitrary degree.

In the case of one-dimensional cellular automaton (CA), a hybrid CA (HCA) is the member whose evolution of the cells is dependent on nonunique global functions. The HCAs exhibit a wide range of traveling and stationary localizations in their evolution. We focus on HCA with memory (HCAM) because they produce a host of gliders and complicated glider collisions by introducing the hybrid mechanism. In particular, we undertake an exhaustive search of gliders and describe their collisions using quantitative approach in HCAM$(43,74)$. By introducing the symbol vector space and exploiting the mathematical definition of HCAM, we present an analytical method of complex asymptotic dynamics of the gliders.

Even though the fundamental main structure of cellular automata (CA) is a discrete special model, the global behaviors at many iterative times and on big scales could be a close, nearly a continuous, model system. CA theory is a very rich and useful phenomena of dynamical model that focuses on the local information being relayed to the neighboring cells to produce CA global behaviors. The mathematical points of the basic model imply the computable values of the mathematical structure of CA. After modeling the CA structure, an important problem is to be able to move forwards and backwards on CA to understand their behaviors in more elegant ways. A possible case is when CA is to be a reversible one. In this paper, we investigate the structure and the reversibility of two-dimensional (2D) finite, linear, triangular von Neumann CA with null boundary case. It is considered on ternary field ${\mathbb{Z}}_3$ (i.e. 3-state). We obtain their transition rule matrices for each special case. For given special triangular information (transition) rule matrices, we prove which triangular linear 2D von Neumann CAs are reversible or not. It is known that the reversibility cases of 2D CA are generally a much challenged problem. In the present study, the reversibility problem of 2D triangular, linear von Neumann CA with null boundary is resolved completely over ternary field. As far as we know, there is no structure and reversibility study of von Neumann 2D linear CA on triangular lattice in the literature. Due to the main CA structures being sufficiently simple to investigate in mathematical ways, and also very complex to obtain in chaotic systems, it is believed that the present construction can be applied to many areas related to these CA using any other transition rules.

In this paper, the chaotic dynamics of composition operators on the space of real-valued continuous functions is investigated. It is proved that the hypercyclicity, topologically mixing property, Devaney chaos, frequent hypercyclicity and the specification property of the composition operator are equivalent to each other and are stronger than dense distributional chaos. Moreover, the composition operator $C_{\varphi }$ exhibits dense Li–Yorke chaos if and only if it is densely distributionally chaotic, if and only if the symbol $\varphi$ admits no fixed points. Finally, the long-time behaviors of the composition operator with affine symbol are classified in detail.

Recently, the gradient method with perturbation (GP) was proposed for metaheuristic methods of solving continuous global optimization problems. Its updating system based on the steepest descent method is chaotic because of its perturbations along the standard basis vectors, which can strengthen the diversification of search. The sufficient condition for its chaoticity was theoretically shown, which implies that two kinds of influence degrees of the perturbations in the updating system should be greater than some constants. In this paper, we extend the updating system of the GP into a more general one for metaheuristic methods, which does not necessarily require the descent direction of the objective function, and which can have perturbations along appropriate orthogonal basis vectors for each problem. Furthermore, since the condition for the chaoticity shown in the previous work is too restricted for practical use, we theoretically show a weaker sufficient condition for the extended system, which is derived by varying the constant lower bounds for the two kinds of influence degrees.

This paper deals with the design-oriented analysis of slow-scale bifurcations in single phase DC–AC inverters. Since DC–AC inverter belongs to a class of nonautonomous piecewise systems with periodic equilibrium orbits, the original averaged model has to be translated into an equivalent autonomous one via a virtual rotating coordinate transformation in order to simplify the theoretical analysis. Based on the virtual equivalent model, eigenvalue sensitivity is used to estimate the effect of the important parameters on the system stability. Furthermore, theoretical analysis is performed to identify slow-scale bifurcation behaviors by judging in what way the eigenvalue loci of the Jacobian matrix move under the variation of some important parameters. In particular, the underlying mechanism of the slow-scale unstable phenomenon is uncovered and discussed thoroughly. In addition, some behavior boundaries are given in the parameter space, which are suitable for optimizing the circuit design. Finally, physical experiments are performed to verify the above theoretical results.

Pyragas time-delayed feedback control has proven itself as an effective tool to noninvasively stabilize periodic solutions. In a number of publications, this method was adapted to equivariant settings. In this paper, we consider $O_4$-symmetric systems of van der Pol and optical oscillators coupled in a cube-like configuration. These systems undergo equivariant Hopf bifurcations giving rise to multiple branches of unstable periodic solutions. We introduce a delayed control term, which ensures stabilization of a selected branch. Group theoretic restrictions which help to shape our choice of control are discussed. Furthermore, we explicitly describe the domains in a two-dimensional parameter space for which the periodic solutions of the delayed system are stable.

The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton–zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored.

The continuous increment in the performance of classical computers has been driven to its limit. New ways are studied to avoid this oncoming bottleneck and many answers can be found. An example is the Belousov–Zhabotinsky (BZ) reaction which includes some fundamental and essential characteristics that attract chemists, biologists, and computer scientists. Interaction of excitation wave-fronts in BZ system, can be interpreted in terms of logical gates and applied in the design of unconventional hardware components. Logic gates and other more complicated components have been already proposed using different topologies and particular characteristics. In this study, the inherent parallelism and simplicity of Cellular Automata (CAs) modeling is combined with an Oregonator model of light-sensitive version of BZ reaction. The resulting parallel and computationally-inexpensive model has the ability to simulate a topology that can be considered as a one-bit full adder digital component towards the design of an Arithmetic Logic Unit (ALU).

Lorenz studied the coupled Rosby waves and gravity waves using the differential system

A memristor is the fourth circuit element, which has wide applications in chaos generation. In this paper, a four-dimensional hyperchaotic jerk system based on a memristor is proposed, where the scroll number of the memristive jerk system is controllable. The new system is constructed by introducing one extra flux-controlled memristor into three-dimensional multiscroll jerk system. We can get different scroll attractors by varying the strength of memristor in this system without changing the circuit structure. Such a method for controlling the scroll number without changing the circuit structure is very important in designing the modern circuits and systems. The new memristive jerk system can exhibit a hyperchaotic attractor, which has more complex dynamic behavior. Furthermore, coexisting attractors are observed in the system. Phase portraits, dissipativity, equilibria, Lyapunov exponents and bifurcation diagrams are analyzed. Finally, the circuit implementation is carried out to verify the new system.

We investigate the dynamics of two types of nonautonomous ordinary differential equations with quasi-periodic time-varying coefficients and nonlinear terms. The vector fields for the nonautonomous systems are written as $ẇ=a{\mathrm{\Phi }}_1(w)+b{\mathrm{\Phi }}_2(t,w)$, $w\in {ℝ}^3$, where $a{\mathrm{\Phi }}_1$ is the spacial part and $b{\mathrm{\Phi }}_2$ is the time-varying part, and $a$ and $b$ are real parameters. The first type has a polynomial as the nonlinear term, another type has a continuous periodic function as the nonlinear term. The polynomials and periodic functions have simple zeros. Several examples with numerical experiments are given. It is found by numerical calculation that there might exist only one attractor for the systems with polynomials as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$, and there might exist infinitely many attractors for systems with periodic functions as nonlinear terms and $\left|b\right|\gg max\{1,|a|\}$. For $\left|b\right|$ sufficiently small, the parameter regions for $(a,b)$ are roughly divided into three parts: the spacial region ($\left|a\right|\gg \left|b\right|$), the balance region ($\left|a\right|\approx \left|b\right|$), and the time-varying region ($\left|a\right|\ll \left|b\right|$); (i) for $\left|a\right|\gg \left|b\right|$, the orbits approach some planes depending on the zeros of the polynomials or the periodic functions; (ii) for $\left|a\right|\approx \left|b\right|$, there exist attractors with the number no less than the number of zeros of the polynomials or the periodic functions, implying the existence of infinitely many attractors for systems with periodic functions as nonlinear terms; (iii) for $\left|a\right|\ll \left|b\right|$, the orbits wind around some region depending on the choice of the initial position. The shape of the attractors might be strange or regular for different parameters, and we obtain the existence of ball-like (regular) attractors, two-wings (strange) attractors, and other attractors with different shapes. The Lyapunov exponents are negative. These results reveal an intrinsic relationship between the existence of attractors (or strange dynamics) and the parameters $a$ and $b$ for nonautonomous systems with quasi-periodic coefficients. These results will be very useful in the understanding of the dynamics of general nonautonomous systems, nonautonomous control theory and other related fields.

Multi-scroll Unstable Dissipative Systems (UDS) in $R^3$ which consist of piecewise linear systems are implemented electronically by means of analog computing. The scrolling behavior of the systems can be designed to oscillate along a specific axis or into space depending on the unstable and stable manifolds. In order for a multi-scroll attractor, this switching system must present at least two unstable hyperbolic focus-saddle equilibria with the same stability index, a negative real eigenvalue and a pair of complex conjugated eigenvalues with positive real part. Then, to displace the scrolls among the axes and space different switching control laws must be designed. By taking into consideration the mathematical expressions of the switching systems, the electronic implementations are carried out by means of operational amplifiers representing the real analog physical solution of the systems, from which the voltage is measured representing the states solution.

Control system is an inevitable element of any industrial installation. Its quality affects overall process performance significantly. The assessment, whether control system needs any improvement or not, requires relevant and constructive measures. There are various methods, like time domain based, Minimum Variance, Gaussian and non-Gaussian statistical factors, fractal and entropy indexes. Majority of approaches use time series of control variables. They are able to cover many phenomena. But process complexities and human interventions cause effects that are hardly visible for standard measures. It is shown that the signals originating from industrial installations have multifractal properties and such an analysis may extend standard approach to further observations. The work is based on industrial and simulation data. The analysis delivers additional insight into the properties of control system and the process. It helps to discover internal dependencies and human factors, which are hardly detectable.

We consider a system of three nonidentical coupled phase oscillators in a ring topology. We explore the conditions that must be satisfied in order to obtain the phases at the transition to a synchrony state. These conditions lead to the correct mathematical expressions of phases that aid to find a simple analytic formula for critical coupling when the oscillators transit to a synchronization state having a common frequency value. The finding of a simple expression for the critical coupling allows us to perform a linear stability analysis at the transition to the synchronization stage. The obtained analytic forms of the eigenvalues show that the three coupled phase oscillators with periodic boundary conditions transit to a synchrony state when a saddle-node bifurcation occurs.

This paper deals with modeling and bifurcation analysis of two-stage Boost inverters. Since the effect of the nonlinear interactions between source-stage converter and load-stage inverter causes the “hidden” second-harmonic current at the input of the downstream H-bridge inverter, an observer-pattern modeling method is proposed by removing time variance originating from both fundamental frequency and hidden second harmonics in the derived averaged equations. Based on the proposed observer-pattern model, the underlying mechanism of slow-scale instability behavior is uncovered with the help of eigenvalue analysis method. Then eigenvalue sensitivity analysis is used to select some key system parameters of two-stage Boost inverter, and some behavior boundaries are given to provide some design-oriented information for optimizing the circuit. Finally, these theoretical results are verified by numerical simulations and circuit experiment.

Wang–Chen system with only one stable equilibrium as well as the coexistence of hidden attractors has attracted increasing interest due to its striking features. In this work, the effect of state feedback on Wang–Chen system is investigated by introducing a further state variable. It is worth noting that a new chaotic system without equilibrium is obtained. We believe that the system is an interesting example to illustrate the conversion of hidden attractors with one stable equilibrium to hidden attractors without equilibrium.