International Journal of Bifurcation and Chaos

Equilibria are a class of attractors that host inherent stability in a dynamic system. Infinite number of equilibria and chaos sometimes coexist in a system with some connections. Hidden chaotic attractors exist independent of any equilibria rather than being excited by them. However, the equilibria can modify, distort, eliminate, or even instead coexist with the chaotic attractor depending on the distance between the equilibria and chaotic attractor. In this paper, chaotic systems with infinitely many equilibria are considered and explored. Extra surfaces of equilibria are introduced into the chaotic flows, showing that a chaotic system can maintain its basic dynamics if the newly added equilibria do not intersect the original attractor. The offset-boostable plane of equilibria rescales the frequency of the chaotic oscillation with an almost linearly modified largest Lyapunov exponent or conversely drives the system into periodic oscillation, even ending in a divergent state. Furthermore, additional infinite number of equilibria or even a solid space of equilibria are safely nested into the chaotic system without destroying the original dynamics, which provides an alternate permanent location for a dynamical system. A circuit simulation agrees with the numerical calculation.

The phenomena of bifurcation and chaos are studied for a class of second order nonlinear nonautonomous ordinary differential equations, which may be formulated by the nonlinear radially symmetric motion of the dynamically loaded hyperelastic spherical membrane composed of the Rivlin–Saunders material model with a noninteger power-law exponent. Firstly, based on the variational principle, the governing equation describing the problem is obtained with the spherically symmetric deformation assumption. Then, the dynamic characteristics of the system are qualitatively analyzed in detail in terms of different values of material parameters. Particularly, for a given constant load, the parameter spaces describing the bifurcation behaviors of equilibrium curves are established and the characteristics of equilibrium points are presented; for a periodically perturbed load, the quasi-periodic and chaotic behaviors are discussed for the systems with two and three equilibrium points, respectively.

Coexisting attractors may arise from many different sources such as hidden basins of attraction or peculiarly organized bifurcation structures. By exploiting the regions of mismatched bifurcations between the system and its fixed points, this study investigates coexisting attractors in a six-dimensional extension of the Lorenz system. This six-dimensional extension takes into account additional physical ingredients, namely, rotation and density-affecting scalar, which are not considered in the original Lorenz system. These newly considered physical ingredients can influence the bifurcation structures and thus the system’s characteristics with regard to coexisting attractors. Once the potential regions of coexisting attractors are identified in the parameter spaces, the coexistence of periodic and point attractors and that of two different periodic orbits in addition to the well-known coexistence of chaos and stability are demonstrated through the solution trajectories and attractor basin boundaries.

In the last few decades, the discrete chaotification of difference equations has gained the considerable attention of academicians and scholars due to its tremendous applications in many branches of science, such as cryptography, traffic control models, secure communications, weather forecasting and engineering. In this article, a modulated logistic system is studied and superior chaos is reported through period-doubling bifurcation, period-three orbit and Lyapunov exponent properties. It is interesting to see that the modulated system admits the superiority in chaos due to the presence of an extra degree of freedom of an ordered parameter $\alpha$. Analytical as well as numerical simulations are demonstrated to examine the period-doubling and Lyapunov exponent properties for some particular values of parameter $\alpha$. Moreover, it is speculated that the superiority in chaos may be used in chaos-based cryptography, in the future.

A new 4D memristive chaotic system with an infinite number of equilibria is proposed via exhaustive computer search. Interestingly, such a new memristive system has a plane of equilibria and two other lines of equilibria. Lyapunov exponent and bifurcation analysis show that this system has chaotic solutions with coexisting attractors. The basins of attraction of the coexisting attractors show chaos, stable fixed-points, and unbounded solutions. Furthermore, the 2D parameter space of the system is explored to find the optimum values of the parameters using the ALO (Ant Lion Optimizer) optimization algorithm.

For two families of planar piecewise smooth polynomial differential systems, whose unperturbed system has a degenerate center at the origin, we study the biggest lower bound for the maximum number of limit cycles bifurcating from the periodic orbits of the center. These results are extensions of the known ones on unperturbed nondegenerate $\mathrm{\Sigma }$-center, derived from a nonsmooth harmonic oscillator model, to degenerate $\mathrm{\Sigma }$-center. Our study involves some new computational treatments. The main tools are the generalized polar coordinate change and the generalized Lyapunov polar coordinate change together with an averaging theory for one-dimensional piecewise smooth differential equations. Finally, we present two Maple programs for computing the averaging functions and consequently the biggest lower bound on the maximum number of limit cycles of degenerate $(2k,2l)$-center under general polynomial perturbations of degree $n$.

We prove the $L^1$-norm and bounded variation norm convergence of a piecewise linear least squares method for the computation of an invariant density of the Foias operator associated with a random map with position dependent probabilities. Then we estimate the convergence rate of this least squares method in the $L^1$-norm and the bounded variation norm, respectively. The numerical results, which demonstrate a higher order accuracy than the linear spline Markov method, support the theoretical analysis.

In this paper, a chaotic communication system based on the combination of Improved–Differential chaos-shift keying (IDCSK) scheme and $2\times 2$ MIMO technique (2T2R), namely 2T2R-IDCSK, is investigated. We point out that there are differences in correlation characteristics between the conventional and time-reversed chaotic sequences. From this aspect, bit-error-rate (BER) performance of the system over an additive white Gaussian noise (AWGN) channel is analyzed taking into account the correlation effect of time-reversed chaotic sequences. BER expressions are verified by Monte-Carlo simulation. The obtained results prove that the 2T2R-IDCSK system performs worse under the correlation effect, however still dramatically outperforms the conventional $2\times 2$ MIMO-DCSK system.

The relationship between currencies in foreign exchange markets has been a topic of significance in economics. Previous studies have focused more on correlations between currencies. However, the detection of causality can reveal their inherent laws. Although the traditional Granger causality test can identify causality, it cannot take into account the nature and intensity of the causality. Thus, the objective of this paper is to identify the causalities of currencies from the perspective of dynamics. In this paper, we select 25 currencies (with the US dollar (USD) as the numeraire) from foreign exchange markets, as they occupy large shares in their regions. To detect the causalities of the foreign exchange markets, we combine PC (pattern causality) theory and complex networks to construct directed and weighted causality networks, in which the nodes represent the currencies and the directed edges represent the causal intensities. Furthermore, we study the symmetry of each causality and quantify the symmetry degree. The results demonstrate that causalities exist between currencies that differ in terms of nature and intensity. The positive causality network exhibits substantial robustness, which can be regarded as the dominant causal relationship in the foreign exchange markets, although a few exceptions are encountered, such as the dominant negative and disordered causalities between currency pairs. In addition, the dominant causalities between most currencies are symmetric in terms of nature, and they also exhibit symmetry in terms of intensity. Furthermore, by gradually deleting the network by thresholding according to the edge weights, we identify the important driving currencies of the markets. This paper may provide valuable information for investors and supervisory departments.

We prove that all Hopf bifurcations in the Nicholson’s blowfly equation are supercritical as we increase the delay. Earlier results treated only the first bifurcation point, and to determine the criticality of the bifurcation, one needed to substitute the parameters into a lengthy formula of the first Lyapunov coefficient. With our result, there is no need for such calculations at any bifurcation point.

An effort to study one-dimensional nonuniform elementary number conserving cellular automata (NCCA) rules from an exponential order rule space of cellular automata is an excellent computational task. To perform this task effectively, a mathematical heritage under the number of conserving functions over binary strings of length $n$ has been highlighted along with their number conserving cellular automata rules (either uniform or nonuniform). A basic approach for the construction of some feasible nonuniform NCCA rules of any finite configuration with the assistance of nine uniform elementary CA rules has been investigated. From our construction procedure, recurrence equations are formulated as suitably solved to ascertain the actual range of NCCA rules. The state transition diagrams (STDs) of NCCA rules are analyzed. While classifying the binary strings through STDs, we found a fascinating optical insight that equal weight strings from a class whose cardinality is the same as the binomial coefficient $C(n,k)$ where $n$ is the length and $k$ is the weight of the binary string.

Hidden symmetries have been previously explored in the context of coupled cell networks and coupled cell systems. These include interior symmetry, quotient symmetry and quotient interior symmetry. We introduce here an equivariant degree theory that incorporates these different forms of hidden symmetry based on lattice structures of synchrony subspaces. The result is a unified theory capable of treating synchrony-breaking Hopf bifurcation problems in coupled cell systems with various forms of hidden symmetries, which leads to full topological classifications of bifurcating branches arising from single bifurcation points under mutual influence of these hidden symmetries.

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is less than the threshold, thus the exploitation is forbidden. However, the exploitation is permitted if the density of the prey population increases and exceeds the threshold. The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically. Moreover, the regions of sliding and crossing segments are analyzed. The dynamic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria are investigated. Numerically, the bifurcation diagram and maximum Lyapunov exponents are computed and plotted to show the complex dynamics of FFCM, for instance, it has stable periodic, double periodic and chaotic solutions as well as double periodic sliding bifurcation. It is demonstrated that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of FFCM.

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

For a family of planar piecewise linear differential systems with two zones both having virtual foci, we investigate the appearance of limit cycles bifurcated from a global center (i.e. Poincaré bifurcations of limit cycles) when the discontinuity boundary is perturbed by the appearance of a nonregular point. Precisely, when the discontinuity boundary, which is a straight line, becomes two rays starting from the same point, we prove that accompanied by the birth of a limit cycle, the center either disappears or becomes local. So, our main results contain some sufficient conditions satisfied by the system parameters that guarantee the appearance of a limit cycle surrounding a period annulus. Moreover, our results provide a new method to design such a system having any number of limit cycles.

It was shown previously that the synergy of the multi-degree-of-freedom (multi-DOF) and the mechanical stopper techniques is an effective way to further broaden the operation bandwidth of vibration energy harvesters. Considering the stochastic factor at the same time, this paper is devoted to developing a theoretical method for analyzing the dynamical characteristics of an impact-engaged 2DOF energy harvesting system under stochastic excitation, in which the impact is modeled as a nonlinear piecewise function. The existence of the impact and multi-DOF brings complexity in analyses, and the main theoretical method utilized in this paper is the stochastic averaging method which is suitable for multi-DOF quasi-nonintegrable systems. After that, the joint and marginal probability density functions of the system are derived, and numerical examples are shown to verify the effectiveness of the theoretical method. It is noteworthy that the theoretical method has the potential to be extended to more complex multi-DOF systems with stoppers. Subsequently, the stochastic response, bifurcations in the stochastic and corresponding deterministic systems, and the relationship between the D-bifurcation and the mean square voltage are discussed. Effects of crucial system coefficients, the impact stiffness and distance on the system are investigated as well.

Interaction among different units in a network of oscillators may often lead to quenching of oscillations and the importance of oscillation quenching can be found in controlling the dynamics of many real world systems. But there are also many real life phenomena where suppression of oscillation should be avoided for maintaining the sustained evolution of the system. In this work, we propose a self-feedback control scheme through which one is able to either achieve quenching or to retrieve the rhythmic behavior in a network of mean-field diffusively coupled systems. It is found that for proper choice of the strength of the control signal, the system converges to an oscillatory state from oscillation quenched state and for further increase of the strength the system enters in chaotic region. Moreover, reversal of the phase of the control signal can induce suppression of oscillation. Thus, the proposed modification offers a better control on the dynamics of the mean-field coupled system. In addition to this, a new transition phenomenon from inhomogeneous limit cycle (IHLC) to homogeneous limit cycle (HLC) through chaotic route has also been found in the modified system.

This paper completes the description of the generalized Lorenz system (GLS) and hyperbolic generalized Lorenz system (HGLS) along with their canonical forms (GLCF, HGLCF), mostly presented earlier, by deriving explicit state transformation formulas to prove the equivalence between GLS and GLCF, as well as between HGLS and HGLCF. Consequently, complete formulations of the generalized Lorenz canonical systems and forms, and their hyperbolic settings, are obtained and presented. Only potentially chaotic systems are classified, which significantly helps clarify the respective canonical forms. To do so, some tools for systems to exclude chaotic behavior are developed, which are interesting in their own right for general dynamical systems theory. The new insight may inspire future investigations of generalized and canonical formulations of some other types of chaotic systems.

We focus on the two-dimensional stable manifold of Chen system. Based on the Chen system, a new system with nonlinear control is put forward, which exhibits brand new dynamical phenomena such as the coexistence of heteroclinic orbits with periodic solutions, or two new attractors. The emergence of manifold surface is found by tangential self-intersection. The two-dimensional unstable manifold of saddle-focus tends to another equilibrium solution asymptotically, the underlying heteroclinic bifurcation and the “twins” unstable manifolds are bulit from two saddle-focus respectively.