International Journal of Bifurcation and Chaos

A memristor with coexisting pinched hysteresis loops and twin local activity domains is presented and analyzed, with an emulator being designed and applied to the classic Chua’s circuit to replace the diode. The memristive system is modeled with four coupled first-order autonomous differential equations, which has three equilibria determined by three static equilibria of the memristor but not controlled by the system parameters. The complex dynamics of the system are analyzed by using compound coexisting bifurcation diagrams, Lyapunov exponent spectra and phase portraits, including point attractors, limit cycles, symmetrical chaotic attractors and their blasting, extreme multistability, state-switching without parameter, and transient chaos. Of particular surprise is that the extreme multistability of the system is hidden and symmetrically distributed. It is found that the existence of transient chaos in the specified parameter domain is determined by using bifurcation diagrams within different time durations and Lyapunov exponents with chaotic sequences. Finally, the symmetrical chaotic attractor and the system blasting are verified by digital signal processing experiments, which are consistent with the numerical analysis.

We are concerned with the Turing instability and pattern caused by cross-diffusion in a strongly coupled spatial predator–prey system. We explore how cross-diffusion destabilizes the spatially uniform steady state which is stable in reaction–diffusion systems, and explicitly describe the Turing space under certain conditions. Particularly, when the parameter values are taken in the Turing–Hopf domain, in which the spatiotemporal dynamical behavior is influenced by both Hopf and Turing instabilities, we investigate the formation of all possible patterns, including non-Turing structures such as wave pattern, competing dynamics as well as stationary Turing pattern. Furthermore, numerical simulations are illustrated to verify our theoretical findings.

Following the scheme inspired by Tsallis [Jagannathan & Sudeshna, 2005; Patidar & Sud, 2009], we study the Gaussian map and its $q$-deformed version. We compute the topological entropies of the discrete dynamical systems which are obtained for both maps, the original Gaussian map and its $q$-modification. In particular, we are able to obtain the parametric region in which the topological entropy is positive. The analysis of the sign of Schwarzian derivative and the topological entropy allow us a deeper analysis of the dynamics. We also highlight the coexistence of attractors, even if it is possible to determine a wide range of parameters in which one of them is a chaotic attractor.

The ionization of hydrogen in a circularly polarized electric field consists of the electron escaping over a rotating Stark saddle. The escape is mediated by the stable and unstable manifolds of a normally hyperbolic invariant manifold associated with the saddle. General principles guarantee the existence of this normally hyperbolic invariant manifold which is referred to as a transition state for energies close to the saddle. We use a numerical procedure to continue the transition state to high energies and study the dynamics restricted to the transition state. We find that for the strength of the electric field under consideration, the transition state persists to very high energies and the restricted dynamics remains amazingly regular. Our analysis moreover includes a study of homoclinic orbits connecting the transition state to itself.

This study investigates the behavior of a damped, inelastic, sinusoidally forced impact oscillator which has its barrier placed such that the oscillator always vibrates under compression about its subharmonic resonant frequencies. The Poincaré sections at near subharmonic resonance conditions exhibit finger-shaped chaotic attractors, similar to the strange attractor mapping of Hénon and the ones found by Holmes in his study of chaotic resonances of a buckled beam. The number of such fingers are observed to increase as the barrier distance from the equilibrium is decreased. These chaotic states are interspersed with regimes of periodic behavior, with the periodicity being in accordance with well defined period adding laws. This study also focuses on the ordered behavior of the one-impact period-$p$ orbits around the higher subharmonics of the oscillator.

The generalized Hamiltonian function is proposed for the brushless DC motor (BLDCM) chaotic system. The Hamiltonian and Casimir functions are derived from the generalized Hamiltonian function. In this way the Casimir energy is proven to be a special type of the generalized Hamiltonian function. The derivative of the Hamiltonian function is used for analyzing the various dynamical behaviors under different combination of energy components. An analytical optimal bound of the BLDCM is simply proposed from the Hamiltonian power. Along the study, the comparison between the Hamiltonian and Casimir powers is conducted, and physical interpretations and mechanism revealing the onset of chaos are provided for the BLDCM chaotic system. Bifurcation analysis through the Hamiltonian power and Casimir power identifies the different dynamic patterns.

In this paper, a prey–predator-top predator food chain model with nonmonotonic functional response in the predators is studied. With an emphasis on the nutrition conversion rate of predator to top predator, one can get two important thresholds: the top predator extinction threshold and the coexistence threshold. The top predator will die out if the nutrition conversion rate of predator to top predator is less than the top predator extinction threshold; the prey, predator and top predator will coexist if the rate is larger than the coexistence threshold. While between the two thresholds is a bistable interval. When the nutrition conversion rate of predator to top predator is in the bistable interval, the system will see the emergence of bistability. The bifurcation analysis of the system depending on parameters indicates that it exhibits saddle-node bifurcation and Hopf bifurcation phenomena.

In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results.

In this paper, we study a class of $Z_p$-equivariant planar polynomial differential systems $ż=(a+\mathrm{\text{i}})z+(b+\mathrm{\text{i}})z|z{|}^{2(p-2)}+cz|z{|}^2-\frac{5\mathrm{\text{i}}}{2}{\overline{z}}^{p-1}$. It is shown that for any $4\le p\in \mathbb{N}$ there is a differential system of the above type having at least $2p$ limit cycles. This is proved by estimating the number of zeros of the first-order Melnikov function.

One of the main applications for chaotic circuits is the production of aperiodic signals with many of the characteristics of noise for secure communications and similar uses. However, the probability distribution function (pdf) of such signals is usually far from Gaussian. This paper describes a new chaotic circuit based on the recently proposed signum thermostat that produces signals whose pdf is accurately Gaussian. Data from the constructed circuit are analyzed and shown to be in agreement with the theoretical prediction.

In this work, by using an algebraic criterion presented by us in an earlier paper, we determine the conditions on the parameters in order to guarantee the nonchaotic behavior for some classes of nonlinear third-order ordinary differential equations of the form $\stackrel{\mathrm{...}}{x}=j(x,ẋ,ẍ),$ called jerk equations, where $j$ is a polynomial of degree $n$. This kind of equation is often used in literature to study chaotic dynamics, due to its simple form and because it appears as mathematical model in several applied problems. Hence, it is an important matter to determine when it is chaotic and also nonchaotic. The results stated here, which are proved using the mentioned algebraic criterion, corroborate and extend some results already presented in literature, providing simpler proofs for the nonchaotic behavior of certain jerk equations. The algebraic criterion proved by us is quite general and can be used to study nonchaotic behavior of other types of ordinary differential equations.

Driver fatigue has caused numerous vehicle crashes and traffic injuries. Exploring the fatigue mechanism and detecting fatigue state are of great significance for preventing traffic accidents, and further lessening economic and societal loss. Due to the objectivity of EEG signals and the availability of EEG acquisition equipment, EEG-based fatigue detection task has raised great attention in recent years. Although there exist various methods for this task, the study of fatigue mechanism and detection of fatigue state still remain much to be explored. To investigate these problems, a multivariate weighted ordinal pattern transition (MWOPT) network is proposed in this paper. To be specific, a simulated driving experiment was first conducted to obtain the EEG signals of subjects in alert state and fatigue state respectively. Then the MWOPT network is constructed based on a novel Shannon entropy. To probe into the mechanism underlying fatigue behavior, the small-worldness index is extracted from the generated MWOPT network. Furthermore, the nodal degree index is input into a classifier to distinguish the fatigue state from alert state. The obtained high accuracy indicates the effectiveness of the proposed network for EEG-based fatigue detection. Besides, four nodes are found to play an important role in identifying fatigue state. These results suggest that the proposed method enables to analyze nonlinear multivariate time series and investigate the driving fatigue behavior.

This paper investigates the problem of searching for the robust initial values of noisy fractional-order chaotic systems when the desired output is given. The problem is addressed under the framework of robust evolutionary optimization. Two different ways of adding noise are considered: (1) the noise is added to the initial point; (2) the noise is added to the orbit of the system. A series of experiments are conducted to validate the effectiveness of robust evolutionary optimization. The experimental results reveal that robust initial values of noisy fractional-order chaotic systems can be obtained in an efficient way by introducing robust evolutionary optimization.

Re-emergence of cholera threatens people’s health globally. However, its periodic re-emerging outbreaks are still poorly understood. In this paper, we develop a simple ordinary differential equation (ODE) model to study the cholera outbreak cycles. Our model involves both direct (i.e. human-to-human) and indirect (i.e. environment-to-human) transmission routes, due to the multiple interactions between the human host, the pathogen, and the environment. In particular, we model the pathogen searching distance as a Poisson point process, and then formulate the host-pathogen encounter (HPE) rate. A thorough mathematical analysis is performed to investigate local and global dynamics of the model. Necessary and sufficient condition under which the backward bifurcation occurs is derived. Fold, Hopf, and Bogdanov–Takens bifurcations are studied with original model parameter values to reveal their relations with model behaviors. One- and two-dimensional bifurcation diagrams are provided to categorize model dynamics with respect to its parameter values. Analytical and numerical analyses show that our simple model is sufficient to exhibit complex epidemic patterns of cholera dynamics including bistability and annual and multiannual periodic outbreaks. Our result regarding the backward bifurcation and complex dynamics of cholera epidemics highlight the challenges in the prevention and control of the disease.

In this paper, the amplitude equations of a Gray–Scott model without (or with) the feedback time delay are derived based on weakly nonlinear method, by which the selection of Turing patterns for this model can be theoretically determined. As a result, the effects of the diffusion coefficient ratio and the time delay factor on the Turing pattern can be investigated as the main purpose of this paper. If one of the diffusion coefficients is chosen as the bifurcation control parameter in the procedure of the amplitude equation at first, it is proved that the first-order bifurcation of the Turing patterns is only determined by the diffusion coefficient ratio and independent of the concrete value of each diffusion coefficient once the parameters of the reaction terms are fixed as the appropriate constants in the regions of Turing patterns. Furthermore, the feedback time delay factor has no effect on the first-order bifurcation of the Turing patterns, but affects the morphological characteristics of the Turing patterns, especially in the case of large ratio of the diffusion coefficients. With time increasing, the feedback time delay factor can postpone the formation of the Turing patterns and cause the oscillations of Turing patterns at each spatial position. By implementing the numerical calculations for this model, the various Turing patterns with different values of the diffusion coefficient ratios are presented, which really verify the dependence of the diffusion coefficient ratio and independence of the feedback time delay on the first-order bifurcation of the Turing patterns.

Some characteristics of mean sensitivity and Banach mean sensitivity using Furstenberg families and inverse limit dynamical systems are obtained. The iterated invariance of mean sensitivity and Banach mean sensitivity are proved. Applying these results, the notion of mean sensitivity and Banach mean sensitivity is extended to uniform spaces. It is proved that a point-transitive dynamical system in a Hausdorff uniform space is either almost (Banach) mean equicontinuous or (Banach) mean sensitive.

Synchronization in complex networks is an evergreen subject with many practical applications across the natural and social sciences. The stability of synchronization is thereby crucial for determining whether the dynamical behavior is stable or not. The master stability function is commonly used to that effect. In this paper, we study whether there is a relation between the stability of synchronization and the proximity to certain bifurcation types. We consider four different nonlinear dynamical systems, and we determine their master stability functions in dependence on key bifurcation parameters. We also calculate the corresponding bifurcation diagrams. By means of systematic comparisons, we show that, although there are some variations in the master stability functions in dependence on bifurcation proximity and type, there is in fact no general relation between synchronization stability and bifurcation type. This has important implication for the restrained generalizability of findings concerning synchronization in complex networks for one type of node dynamics to others.

In this article using an analytical method called Fishing principle we obtain the region of parameters, where the existence of a homoclinic orbit to a zero saddle equilibrium in the Lorenz-like system is proved. For a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the obtained region of parameters is organized, which leads to the discovery of new bifurcation scenarios.

We investigate the effect of a memristive element on the dynamics of a chaotic system. For this purpose, the chaotic Chua’s oscillator is extended by a memory element in the form of a double-barrier memristive device. The device consists of ${\text{Au/NbO}}_{\mathrm{\text{x}}}$/Al₂O₃/Al/Nb layers and exhibits strong analog-type resistive changes depending on the history of the charge flow. In the obtained system we observe strong changes in the dynamics of chaotic oscillations. The otherwise fluctuating amplitudes of Chua’s system are disrupted by transient silent states. Numerical simulations and analysis of the extended model reveal that the underlying dynamics possesses slow–fast properties due to different timescales between the memory element and the base system. Furthermore, the stabilizing and destabilizing dynamic bifurcations are identified that are traversed by the system during its chaotic behavior.