International Journal of Bifurcation and Chaos

In this work an activator-depleted reaction–diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using Chebyshev nonperiodic grid for the radial variable and Fourier periodic grid for the angular variable to verify the nodal lines and eigen-surfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatiotemporal periodic behavior is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction–diffusion rates.

Anticipated synchronization is a counterintuitive synchronization regime between a master and a slave dynamical system in which there is a negative phase difference between the driver and the driven system. By studying a set of simple neural oscillators, we unveil the dynamical mechanisms required to generate this phenomenon. We study master–slave configurations where the slave system is, when uncoupled, in a quiescent excitable state. We exemplify our results by describing the dynamics of a dynamical system proposed to model the part of a songbird’s brain involved in song production.

This paper deals with stability and local bifurcations of two-dimensional (2D) spatiotemporal discrete systems. Necessary and sufficient conditions for asymptotic stability of the systems are obtained. They prove to be more accurate than those in the current literature. Some definitions for the bifurcations of 2D spatiotemporal discrete systems are also given, and an illustrative example is provided to explain the new results.

The dynamics of a chain of interacting active particles of Rayleigh-type is studied. Particles are interconnected via Morse potential forces. The steady-state modes (attractors) of the chain with periodic boundary conditions look like cnoidal waves with a uniform distribution of the particles’ density maxima along the chain. However, if the system starts from random initial conditions, a metastable state with nonuniform distribution of density maxima is formed. Characteristics of metastable states, excitation probability of different modes and their lifetimes are studied by numerical simulation.

We analyze the dynamics of the Poincaré map associated with the center manifold equations of double-diffusive thermosolutal convection near a codimension-four bifurcation point when the values of the thermal and solute Rayleigh numbers, $R_T$ and $R_S$, are comparable. We find that the bifurcation sequence of the Poincaré map is analogous to that of the (continuous) Lorenz equations. Chaotic solutions are found, and the emergence of strange attractors is shown to occur via three different routes: (1) a discrete Lorenz-like attractor of the three-dimensional Poincaré map of the four-dimensional center manifold equations that forms as the result of a quasi-periodic homoclinic explosion; (2) chaos that follows quasi-periodic intermittency occurring near saddle-node bifurcations of tori; and, (3) chaos that emerges from the destruction of a 2-torus, preceded by frequency locking.

In this paper, we investigate the lateral dynamics of a railway wheelset suspended under a moving car with linear springs and dry friction dampers. Both theoretical and numerical methods are used to complement each other. The car runs on an ideal, straight and perfect track with a constant speed. A nonlinear relation between the creepages and the creep forces is used in this paper. The nonsmoothness of this model is due to the dry friction dampers. The speed is selected as the bifurcation parameter. The one-dimensional bifurcation diagram, which gives a general view of the dynamics of the system, is presented. Both symmetric and asymmetric periodic motions, quasi-periodic motions and chaotic motions are found. In addition to bifurcations that can exist in both smooth and nonsmooth systems, a kind of sliding bifurcations that are unique to nonsmooth systems is found. Bifurcation diagrams, phase portraits, Poincaré sections and Lyapunov exponents are presented to ensure that no contradictory results are given. The influence of the conicity of the wheel tread on the Hopf bifurcation type is examined.

In this paper, we study the Poincaré bifurcation of a nonlinear oscillator of generalized Liénard type by using the Melnikov function. The oscillator has weak damping terms. When the damping terms vanish, the oscillator has a heteroclinic loop connecting a nilpotent cusp to a hyperbolic saddle. Our results reveal that: (i) the oscillator can have at most four limit cycles bifurcating from the corresponding period annulus. (ii) There are some parameters such that three limit cycles emerge in the original periodic orbit domain. (iii) Especially, we give a rigorous proof that $l$ limit cycle(s) can emerge near the original singular loop and $m$ limit cycle(s) can emerge near the original elementary center with $l+m=3$.

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof $1:1:4$ resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

The paper discusses the fractional Rössler system and the dependence of its dynamics on some entry parameters. An explicit algorithm for a priori determination of fractional Hopf bifurcations is derived and scenarios documenting a route of the system from stability to chaos are performed with respect to a varying system’s fractional order as well as to a varying system’s coefficient. Contrary to the existing results, the searched values of the fractional Hopf bifurcations follow directly from a revealed analytical dependence between these two systems’ entries. Their various critical values are established and confirmed by numerical experiments demonstrating not only the loss of stability of an equilibrium point, but also other phenomena of transition to chaotic behavior. In addition, we suggest an active control method for synchronization of two chaotic fractional-order Rössler systems. Our theoretical analysis enables to synchronize them for any value of a free parameter under which the master system displays a chaotic behavior.

The predator–prey model is fundamentally important to study the growth law of the population in nature. In this paper, we propose a diffusive predator–prey model, in which we also consider time delay in the gestation time of predator and Michaelis–Menten type predator harvesting. By analyzing the distribution of eigenvalues, we investigate the stability of the coexisting equilibrium and the existence of Hopf bifurcation using time delay as bifurcation parameter. We analyze the property of Hopf bifurcation, and give an explicit formula for determining the direction and the stability of Hopf bifurcation. Finally, some numerical simulations are given to support our results.

Given an $m$-parameterized family of $n$-dimensional vector fields, with an equilibrium point with linearization of eigenvalue zero with algebraic multiplicity $k$, with $k\le m$, and geometric multiplicity one, our goal in this paper is to find sufficient conditions for the family of vector fields such that the dynamics on the $k$-dimensional $m$-parameterized center manifold around the equilibrium point becomes locally topologically equivalent to a given unfolding. Finally, the result is applied to the study of the Rössler system.

This paper is devoted to the study of stability and bifurcations of homoclinic loops for planar vector fields. For a given homoclinic loop, a sequence of loop numbers can be defined such that the stability and bifurcations of the loop are determined by the first nonzero term of the sequence. Formulas for the first several loop numbers were established in the past. In this paper, we will introduce general formulas for the loop numbers for both the single and double homoclinic loops.

We consider nonautonomous dynamical systems consisting of sequences of continuous surjective maps of a compact metric space $X$. Let ${{ℱ}}_0$, ${{ℱ}}_e$ and ${{ℱ}}_p$ denote the space of systems $F={(f_n)}_{n\ge 1}$, which are uniformly convergent, or equicontinuous, or pointwise convergent (to a continuous map), respectively. We show that for $X=I:=[0,1]$, the generic system in any of the spaces has infinite topological entropy, while, if $X$ is the Cantor set, the generic system in any of the spaces has zero topological entropy. This shows, among others, that the general results of the above type for arbitrary compact space $X$ are difficult if not impossible.

Tipping points have been actively studied in various applications as well as from a mathematical viewpoint. A main technique to theoretically understand early-warning signs for tipping points is to use the framework of fast–slow stochastic differential equations. A key assumption in many arguments for the existence of variance and auto-correlation growth before a tipping point is to use a linearization argument, i.e. the leading-order term governing the deterministic (or drift) part of a stochastic differential equation is linear. This assumption guarantees a local approximation via an Ornstein–Uhlenbeck process in the normally hyperbolic regime before, but sufficiently bounded away from, a bifurcation. In this paper, we generalize the situation to leading-order nonlinear terms for the setting of one fast variable. We work in the quasi-steady regime and prove that the fast variable has a well-defined stationary distribution and we calculate the scaling law for the variance as a bifurcation-induced tipping point is approached. We cross-validate the scaling law numerically. Furthermore, we provide a computational study for the predictability using early-warning signs for leading-order nonlinear terms based upon receiver-operator characteristic curves.

In this paper, based on the two-compartment unidirectionally coupled thalamocortical model network, we investigated the transition dynamics of epileptic seizures, by considering the inhibitory coupling strength from cortical inhibitory interneuronal (IN) population to excitatory pyramidal (PY) neuronal population as the key bifurcation parameter. The results show that in the single compartment thalamocortical model, inner-compartment inhibitory functions of IN can make the system transit from the absence seizures to the tonic oscillations. In the case of two-compartment coupled thalamocortical model network, the inter-compartment inhibitory coupling functions from the first compartment can drive the second compartment to more easily initiate the absence and tonic seizures at the lower inhibitory coupling strengths, respectively. Also, the driven functions can make the amplitudes of these seizures vary irregularly. Detailed investigations reveal that along with the various state transitions, the system consecutively undergoes Hopf bifurcations, fold of cycles bifurcations and torus bifurcations, respectively. In particular, the reinforcing inter-compartment inhibitory coupling function can induce the chaotic dynamics. We highlight the unidirectional coupling functions between two compartments which might give new insights into the propagation and evolution dynamics of epileptic seizures.

This paper proposes a new type of memristor with two distinct stable pinched hysteresis loops and twin symmetrical local activity domains, named as a bistable bi-local active memristor. A detailed and comprehensive analysis of the memristor and its associated oscillator system is carried out to verify its dynamic behaviors based on nonlinear circuit theory and Hopf bifurcation theory. The local-activity domains and the edge-of-chaos domains of the memristor, which are both symmetric with respect to the origin, are confirmed by utilizing the mathematical cogent theory. Finally, the subcritical Hopf bifurcation phenomenon is identified in the subcritical Hopf bifurcation region of the memristor.