International Journal of Bifurcation and Chaos

The quadratic iteration is mapped using a nondistributive real scator algebra in three dimensions. The bound set S has a rich fractal-like boundary. Periodic points on the scalar axis are necessarily surrounded by off axis divergent magnitude points. There is a one-to-one correspondence of this set with the bifurcation diagram of the logistic map. The three-dimensional S set exhibits self-similar 3D copies of the elementary fractal along the negative scalar axis. These 3D copies correspond to the windows amid the chaotic behavior of the logistic map. Nonetheless, the two-dimensional projection becomes identical to the nonfractal quadratic iteration produced with hyperbolic numbers. Two- and three-dimensional renderings are presented to explore some of the features of this set.

The border-collision normal form is a piecewise-linear continuous map on ℝ^N that describes the dynamics near border-collision bifurcations of nonsmooth maps. This paper studies a codimension-three scenario at which the border-collision normal form with N = 2 exhibits infinitely many attracting periodic solutions. In this scenario there is a saddle-type periodic solution with branches of stable and unstable manifolds that are coincident, and an infinite sequence of attracting periodic solutions that converges to an orbit homoclinic to the saddle-type solution. Several important features of the scenario are shown to be universal, and three examples are given. For one of these examples, infinite coexistence is proved directly by explicitly computing periodic solutions in the infinite sequence.

The idea to use environmental energy to power electronic portable devices is becoming very popular in recent years. In fact, the possibility of not relying only on batteries can provide devices longer operating periods in a fully sustainable way. Vibrational kinetic energy is a reliable and widespread environmental energy, that makes it a suitable energy source to exploit. In this paper, we study the electrical response of a bistable system, by using a double-well Duffing oscillator, connected to a circuit through piezoceramic elements and driven by both a low (LF) and a high frequency (HF) forcing, where the HF forcing is the environmental vibration, while the LF is controlled by us. The response amplitude at low-frequency increases, reaches a maximum and then decreases for a certain range of HF forcing. This phenomenon is called vibrational resonance. Finally, we demonstrate that by enhancing the oscillations we can harvest more electric energy. It is important to take into account that by doing so with a forcing induced by us, the amplification effect is highly controllable and easily reproducible.

We investigate the effects of time-periodic intercoupling strength on the spiking regularity of a clustered neuronal network. Inside the clustered neuronal network, each cluster has the same regular structure with each node modeled by a stochastic Hodgkin–Huxley neuronal model. From simulation results, we find that there exist some optimal frequencies of the time-periodic intercoupling strength, at which the spiking regularity of the considered clustered neuronal network becomes higher. When we consider the effects of the amplitude of time-periodic intercoupling strength, we find that the clustered neuronal network could exhibit more regular spiking activity at some intermediate amplitude for higher noise level; while for lower noise level, the spiking regularity of the neuronal network decreases with the amplitude increase. And the obtained results are independent of the cluster number.

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

In this paper, we present a new method for chaos generation in nonautonomous impulsive systems. We prove the presence of chaos in the sense of Li–Yorke by implementing chaotic perturbations. An impulsive Duffing oscillator is used to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Moreover, a procedure to stabilize the unstable periodic solutions is proposed.

This paper reviews pulse-excitation as a technique for inducing chaos and provides a review of circuits that have been designed based on this technique. A clock-driven chaotic van der Pol oscillator is also presented. Vulnerability of this class of chaos generators to power supply attacks is demonstrated.

In this paper, a general autonomous van der Pol–Duffing oscillator is studied. Several issues, such as periodic bifurcations and the dynamical structures of the system are investigated either analytically or numerically. Especially, a phenomenon of hidden attractors is noticed and an algorithm for the location of hidden attractors is given. The obtained results show that hidden attractors exist around chaotic attractors.

In this paper, we consider a species predator–prey model given a reaction–diffusion system. It incorporates the Holling type II functional response and a quadratic intra-predator interaction term. We focus on the qualitative analysis, bifurcation mechanisms and pattern formation. We present the results of numerical experiments in two space dimensions and illustrate the impact of the diffusion on the Turing pattern formation. For this diffusion system, we also observe non-Turing structures such as spiral wave, target pattern and spatiotemporal chaos resulting from the time evolution of these structures.

The chaotic spiking and intermittent chaotic bursting were observed from biological experiment on a neural pacemaker. The intermittent chaotic bursting manifested alternation between a phase of nearly period-3 bursting and another phase of irregular bursting, and exhibited a nonsmooth-like structure in the first return map of interspike interval (ISI) series. The chaotic spiking manifested a smooth structure. The intermittent chaotic bursting and spiking simulated in the Chay model were identified by the dissection of the fast-slow variables. The intermittent chaotic bursting manifested nonsmooth-like structures in the first return map of both ISI and slow variable, while spiking exhibited smooth structures. The intermittent chaotic bursting and spiking manifested different scale laws of the averaged length of periodic phase with respect to the changes of parameter value. The scale law of bursting was similar to those of both type V intermittent chaos generated in a nonsmooth map and type I in a smooth system, while that of spiking resembled type I more than type V. The results revealed the dynamics of intermittent chaotic firing patterns, the differences between the intermittent chaotic bursting and spiking patterns and the relationships to the smooth and nonsmooth-like characteristics, provided a deep understanding of intermittent chaotic firing patterns, and indicated different mechanisms of the adjustment of variables from slow to fast between bursting and spiking patterns.

In this paper, we show that a simple three-dimensional quadratic vector field can have at least seven small-amplitude limit cycles, bifurcating from a Hopf critical point. This result is surprisingly higher than the Bautin's result for quadratic planar vector fields which can only have three small-amplitude limit cycles bifurcating from an elementary focus or an elementary center. The methods used in this paper include computing focus values, and solving multivariate polynomial systems using modular regular chains. In order to obtain higher-order focus values for nonplanar dynamical systems, computationally efficient approaches combined with center manifold computation must be adopted. A recently developed explicit, recursive formula and Maple program for computing the normal form and center manifold of general n-dimensional systems is applied to compute the focus values of the three-dimensional vector field.

In a previous paper, we developed a new way to apply the Homotopy Analysis Method (HAM) in the search for periodic orbits in dynamical systems modeled by ordinary differential equations. This method differs from the original in the heuristic search of the frequencies of the cycles. In this paper, we show that the method can be extended to the search for periodic orbits in delay differential equations. Herein, this methodology is applied twice, firstly in an equation of van der Pol type and secondly in an anharmonic oscillator, both systems with a delayed feedback.

In this paper, the dynamics of Chua's circuit in jerk form is presented. It is proved that two forms of Chua's circuit in jerk dynamics can be derived. The electronic circuit of the simpler dynamics has been implemented. The link between Chua's circuit and the search for simple jerk circuits has been established, therefore, proposing complex dynamics (like that of Chua's circuit) in jerk form.

This letter reports an interesting finding that the parametric Lorenz system and the parametric Chen system "shake hands" at a particular point of their common parameter space, as the time variable t → +∞ in the Lorenz system while t → -∞ in the Chen system. This helps better clarify and understand the relationship between these two closely related but topologically nonequivalent chaotic systems.

A novel four-dimensional continuous-time autonomous hyperchaotic system which has no equilibrium is proposed in this paper. By starting from a third-order chaotic system and introducing a further variable performing state feedback, a four-dimensional system exhibiting hyperchaos is obtained. The basic dynamical properties of this system are investigated, such as equilibria and stability, Lyapunov exponent spectrum, and bifurcation diagrams. Furthermore, synchronization via diffusive coupling or control has been addressed. In the latter, parameter identification and synchronization are performed simultaneously. The circuit realization and experimental results are also presented.

In this paper, we consider a model of nonlinear pulse propagation in optical fibers. By investigating the dynamical behavior and bifurcations of solutions of the traveling wave system of PDE, we derive all possible exact explicit traveling wave solutions under different parameter conditions. These results completed the study of traveling wave solutions for the mentioned model posed by [Lenells, 2009].

This paper is focused on the bifurcation of critical periods from a quartic rigidly isochronous center under any small quartic homogeneous perturbations. By studying the number of zeros of the first several terms in the expansion of the period function in ε, it shows that under any small quartic homogeneous perturbations, up to orders 1 and 2 in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system respectively, and the upper bound can be reached. Up to order 3 in ε, there are at most six critical periods from the periodic orbits of the unperturbed system. Moreover, we consider a family of perturbed systems of this quartic rigidly isochronous center, and obtain that up to any order in ε, there are at most two critical periods bifurcating from the periodic orbits of the unperturbed one, and the upper bound is sharp.