International Journal of Bifurcation and Chaos

This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

Current induced oscillations of a space clamped neuron action potential demonstrates a bifurcation scenario originally encapsulated by the four-dimensional Hodgkin–Huxley equations. These oscillations were subsequently described by the two-dimensional FitzHugh–Nagumo Equations in close agreement with the Hodgkin–Huxley theory. It is shown that the FitzHugh–Nagumo equations can to close approximation be reduced to a generalized van der Pol oscillator externally driven by the current. The current functions as an external constant force driving the action potential. As a consequence approximate analytic expressions are derived which predict the bifurcation scenario, the amplitudes of the oscillations and the oscillation periods in terms of the current and the physiological constants of the FitzHugh–Nagumo model. A second reduction permits explicit analytic solution and results in a spiking model which can be multiply coupled and extended to include the dynamics of phase locking, entrainment and chaos characteristic of time-dependent synaptic inputs.

Important components of neural networks are input synapses, action potential generators and output synapses. Rather than modeling a whole neuron in terms of a few ionic channels or as having Hodgkin–Huxley, Morris–Lecar or FitzHugh–Nagumo dynamics, we describe a neuron's action potential generator (APG). An APG may be at the hillock region at the base of an axon or another specific region of a cell. We model it using bifurcation theory based on observations by A. F. Hodgkin about membrane excitability. The result is a simplified model that leads us to view a neural network as comprising input and output synapses (electrical or chemical) that network APGs. These centers of activity are coupled by transfer functions from input synapses to an APG and from an APG to output synapses. The transfer functions account for time delays and signal attenuation that result from input and output structures. While this falls far short of a complete biophysical model of specific neurons in a network, it is consistent with empirical data, it is easily formulated, it is analytically tractable, and computer simulations based on it are straightforward. One outcome is a precise description of the cumulative distribution function (CDF) of action potentials. Since records of cell firing amount to collections of CDFs, the model is for a variable that is accessible to experimental observation. This methodology is applied here to describe bursting neural circuits and embedded loop networks similar to those occurring in basal ganglia.

We studied multistable oscillatory states of a small neural network model and switching of an oscillatory mode. In the present neural network model, two pacemaker neurons are reciprocally inhibited with conduction delay; one pacemaker neuron inhibits the other via an inhibitory nonpacemaker interneuron, and vice versa. The small network model shows bifurcations from quasi-periodic oscillation to chaos via period 3 with increase in the synaptic weight of the reciprocal inhibition. The route to chaos in the network model is different from that in the single pacemaker neuron. The network model exhibits several multistable states. In a regime of a weak inhibitory connection, in-phase beat, out-of-phase beat (period 3), and chaotic oscillation coexist at the multistable state. We can switch an oscillatory mode by an excitatory synaptic input to one of the pacemaker neurons through an afferent path. In a strong inhibitory connection regime, in-phase beat and out-of-phase beat (period 4) coexist at the multistable state. An excitatory synaptic input through the afferent path leads to the transition from the in-phase beat to the out-of-phase beat. The transition from the out-of-phase beat to the in-phase beat is induced by an inhibitory synaptic input via interneurons. A conduction delay, furthermore, causes the spontaneous transition from the in-phase beat to the out-of-phase beat. These transitions can be explained by phase response curves.

The dynamics of nonlinear reaction–diffusion systems is dominated by the onset of patterns, and Fisher equation is considered to be a prototype of such diffusive equations. Here we investigate the integrability properties of a generalized Fisher equation in both (1+1) and (2+1) dimensions. A Painlevé singularity structure analysis singles out a special case (m=2) as integrable. More interestingly, a Bäcklund transformation is shown to give rise to a linearizing transformation for the integrable case. A Lie symmetry analysis again separates out the same m=2 case as the integrable one and hence we report several physically interesting solutions via similarity reductions. Thus we give a group theoretical interpretation for the system under study. Explicit and numerical solutions for specific cases of nonintegrable systems are also given. In particular, the system is found to exhibit different types of traveling wave solutions and patterns, static structures and localized structures. Besides the Lie symmetry analysis, nonclassical and generalized conditional symmetry analysis are also carried out.

The present work describes a family of polynomial noninvertible maps of the plane shared within two open regions: (i) (denoted by Z₀) each point having no real preimage, and (ii) (denoted by Z₂) each point having two real preimages. The regions Z₀, Z₂ are separated by the critical curve LC, locus of points having two coincident preimages. Z₂ is made up of two sheets (giving rise to a plane foliation) joining on LC, each being associated with a well-defined inverse of the map. The considered maps family is structurally unstable. For a wide choice of the parameter space it generates a singular foliation in the sense that the region Z₂ is separated into two zones, Z'₂ and , inside which the two preimages do not have the same qualitative behavior. Moreover, the boundary between Z'₂ and is made up of points having only one real preimage at finite distance, the second one being at infinity. This situation gives rise to a nonclassical homoclinic bifurcation. The maps of the family have another important feature: their inverses present a denominator which vanish along a line of the plane. This has a great consequence on a chaotic attractor structure, when it exists. The imbedding of the map into a wider structurally stable family, generating regions Z_k (k=1,3 being the number of real preimages), permits to understand the foliation nature when the imbedding parameter cancels leading to the structurally unstable map.

The dynamical behavior of nonlinear maps undergoing degenerate period doubling or degenerate Hopf bifurcations is studied via a frequency-domain approach. The technique is based on a discrete-time feedback representation of the system and the application of the well-known engineering tools of harmonic balance to approximate the emerging solutions. More specifically, the results are a higher-order extension of the previous developments obtained by the authors for nondegenerate bifurcations. Two examples are included for illustration.

In this paper we consider the dynamics of the sine circle map with quasiperiodic forcing and very small forcing frequency. One of our goals is to compare the dynamics of our system with those of the system with larger values of the frequency. In particular we discovered an unusual fine structure of the mode-locking regions which disappears with increase in the forcing frequency. We investigated how this fine structure depends on the system parameters and described the bifurcations which occur on the boundaries of the mode-locking regions.

We propose a scheme to detect signals immersed in strong, externally imposed, undesirable noise (jamming) by making use of the principle of stochastic resonance. The strategy is to construct an array of simple nonlinear oscillators and to apply independent, modulating noise to each oscillator. We show that the collective effect of all oscillators and the interplay between nonlinearity and different noise sources can enhance the detectability of the original signal. For proof of principle we focus on periodic signals mixed with in-band, Gaussian jamming. In particular, we show by both extensive numerical computations and physical arguments that measures of the detectability such as the signal-to-noise ratio can be increased in our scheme. We suggest how the scheme can be implemented in laboratory experiments.

Combining the backstepping design and the variable structure control, we propose a robust nonlinear feedback control method to control an uncertain van der Pol oscillator even if there exist system uncertainties and external disturbances in this oscillator. If system uncertainties are estimated and some parameters are chosen suitably, the output of van der Pol osicllator can track arbitrary smooth reference signal. Theoretical analysis and numerical simulations verify the effectiveness of this method.

A general explicit formula is derived for controlling bifurcations using nonlinear state feedback. This method does not increase the dimension of the system, and can be used to either delay (or eliminate) existing bifurcations or change the stability of bifurcation solutions. The method is then employed for Hopf bifurcation control. The Lorenz equation and Rössler system are used to illustrate the application of the approach. It is shown that a simple control can be obtained to simultaneously stabilize two symmetrical equilibria of the Lorenz system, and keep the symmetry of Hopf bifurcations from the equilibria. For the Rössler system, a control is also obtained to simultaneously stabilize two nonsymmetric equilibria and meanwhile stabilize possible Hopf bifurcations from the equilibria. Computer simulation results are presented to confirm the analytical predictions.

In this paper, boundary anticontrol of spatiotemporal chaos for 1D hyperbolic equations is studied. Firstly, a new definition of chaotic vibrations for PDEs is given in terms of the growth rate of the total variations of the solutions with respect to the spatial variable as t→∞. Then, a boundary feedback controller is designed as composing with a sawtooth function, which can drive the originally nonchaotic linear or nonlinear dynamical system chaotic. Finally, as applications, anticontrol of chaos for 1D linear wave equations with linear or nonlinear boundary conditions is discussed.

In this paper, we propose a Star cellular neural network (Star CNN) for associative and dynamic memories. A Star CNN consists of local oscillators and a central system. All oscillators are connected to a central system in the shape of a Star, and communicate with each other through a central system. A Star CNN can store and retrieve given patterns in the form of synchronized chaotic states with appropriate phase relations between the oscillators (associative memories). Furthermore, the output pattern can occasionally travel around the stored patterns, their reverse patterns, and new relevant patterns which are called spurious patterns (dynamic memories).

This paper focuses on the dynamics of coupled Chua's circuits driven by two sinusoidal signals. In particular, it is shown that the application of signals with slightly different frequencies enables the new phenomenon of chaotic beats to be generated. Finally, the application of signals with equal frequencies is discussed, with the aim of understanding the formation of beats in nonlinear circuits.

In this work, the existence of curves of homoclinic connections that bi-spiral around a T-point between two saddle-focus equilibria is detected in Chua's equation. That is, the homoclinic curve emerges spiraling from a T-point in a parameter bifurcation plane and ends, by a different spiral, at the same T-point. This new phenomenon is related to the existence of more than one intersection between the two-dimensional manifolds of the involved equilibria at the T-point.

This Letter reports a simple smooth quadratic autonomous chaotic system with four unstable equilibrium points and three homoclinic orbits, and can display a 3-layer attractor among some others. It is the first example of a simple smooth quadratic system that we know, which can exhibit Šhilnikov chaos without numerical evidence of a route to chaos through period-doubling bifurcations.

We report our results on nonperiodic experimental time series of pressure in a spark ignition engine. The experiments were performed for a low rotational velocity of a crankshaft and a relatively large spark advance angle. We show that the combustion process has many chaotic features. Surprisingly, the reconstructed attractor has a characteristic butterfly shape similar to a chaotic attractor of Lorentz type. The suitable recurrence plot shows that the dynamics of the combustion is a nonlinear multidimensional process mediated by stochastic noise.

The paper addresses robustness of complete stability with respect to perturbations of the interconnections of nominal symmetric neural networks. The influence of the maximum neuron activation gain on complete stability robustness is discussed for a class of third-order neural networks. It is shown that high values of the gain lead to an extremely small complete stability margin of all nominal symmetric neural networks, thus allowing to conclude that complete stability robustness cannot be, in general, guaranteed.

Various bifurcation scenarios from period-1 bursting to period-1 spiking via a complex procedure were simulated in previous theoretical studies on neuronal models. The results revealed a general principle of neuronal firing pattern transitions. In this letter, three types of bifurcation scenarios with respect to extracellar calcium concentration ([Ca⁺⁺]_o) were discovered in experiments on neural pacemakers. Such a series of bifurcation scenarios implied complex structure of bifurcations in the firing pattern transitions of neurons. In the two-dimensional parameter space of Chay model, three classical kinds of bifurcation scenarios with respect to the bifurcation parameter v_c (the reverse potential of calcium concentration) were simulated. By the variation of the conditional parameter, λ_n, the relationship among the three bifurcation scenarios was revealed. The results not only verified the existence of different bifurcation scenarios in real neuronal system, but also indicated that the differences among the bifurcation scenarios were caused by the different configuration of parameters. The physiological significance of such bifurcation structures to a deeper understanding of neural coding mechanism was also discussed.

We consider an experimental setup, modeling the FitzHugh–Nagumo equation without recovery term and composed of a nonlinear electrical network made up of discrete bistable cells, resistively coupled. In the first place, we study experimentally the propagation of topological fronts in the continuum limit where the analytical solution can be obtained. We show that experimental results match the theoretical predictions. The discrete case is then investigated theoretically and in the lattice, emphasizing the pinning of traveling waves.

Multiple Hopf bifurcation in problems with O(2) symmetry leads to the standing and the traveling wave solutions. Swapping the branches at this point is considered by studying an O(2) symmetric problem on . The torus-doubling cascade bifurcations are investigated using canonical coordinate transformation. It is shown that direction reversing chaos can be obtained through a symmetry-increasing bifurcation.

This paper focuses on the analytic and numerical dissipativity of θ-method for delay differential equations with a bounded variable lag. A generalized Halanay inequality is derived, and a sufficient condition is presented to ensure that delay differential equations with a bounded variable lag are dissipative. We then apply θ-method to such delay differential equations, and investigate the numerical dissipativity of the θ-method.

This paper studies a foodweb model concerning hare–lynx-vegetation, and presents a computer assisted proof for existence of horseshoe for Poincaré map derived from this model.

In this paper, a one-parameter Hamiltonian system under cubic perturbations is investigated and the upper bound of the number of zeros of the Abelian integral is obtained by using Horozov and Iliev's method.

The TCP end-to-end congestion control plus RED router queue management can be modeled as a discrete-time dynamical system, which can create complex bifurcating and chaotic behavior. Based on the basic features of the TCP-RED model, we investigate the possibility of controlling bifurcation and chaos in the system via several time-delayed feedback control strategies. Two adaptive parameter-tuning algorithms are proposed and evaluated.

A control scheme is applied between two different oscillators to study their phase synchronization. It utilizes unidirectional signal coupling and only measures the time interval when the trajectories to the two oscillators' attractors cross the Poincaré surfaces respectively. By using this scheme, phase synchronization (without 2π phase slips) can be obtained between two different chaotic systems whose signal variables have large amplitude mismatch. This unidirectional signal coupling also provides a minimum information flow from the driving system to the response system. Therefore it can be used in synchronizing systems with substantially different dynamics via a channel with low information rate.

In this paper, we apply OPCL control to discrete system, and based on relative nonlinear measure, give an algorithm for estimating the radius of stable basin. We rigorously prove that this basin is bound to be of existence for nonlinear discrete system, whose goal dynamics is either periodic orbits or fixed point. We also, in particular, investigate the stable basin in a quadratic polynomial map system, and present that the stable basin is irrelevant to the goal orbits with a negative Jacobian gain matrix. Furthermore, we take the well-known Hénon system and Ikeda system as examples to illustrate the implementation of our theory, and give the corresponding simulations to reinforce our method.