International Journal of Bifurcation and Chaos

The general Euler-Poisson problem of rigid body motion is investigated. We study the three-dimensional algebraic level surfaces of the first integrals, and their topological bifurcations. The main result of this article is an analytical and qualitatively complete description of the projections of these integral manifolds to the body-fixed space of angular velocities. We classify the possible types of these invariant sets and analyze the dependence of their topology on the parameters of the body and the constants of the first integrals. Particular emphasis is given to the enveloping surfaces of the sets of admissible angular velocities. Their pre-images in the reduced phase space induce a Heegaard splitting which lends itself for a general choice of complete Poincaré surfaces of section, irrespective of whether or not the system is integrable.

In this paper the global patterns that result from local interactions between players on a two-dimensional lattice are studied. The assumptions on interaction between players are based on the Prisoner's Dilemma game that has been used extensively in game theory and in the study of biological systems. Each player is located on a square lattice, and is assumed to cooperate or defect, based on mimicking the neighbor with the highest cumulative score from the preceding round of play. The edges of the lattice are glued to form a torus. Computer simulations are conducted for different sized lattices, different payoff values, and different initial conditions. Though the paper is primarily concerned with player behavior without self-interaction, some results with self-interaction are also included. The influence of "ideal" cooperators on the evolution of the system dynamics is also studied. Three generic regimes of behavior are identified. Complex global patterns with complicated dynamics and sometimes unpredictable results occur. Steady-state solutions, simple and complex periodic solutions, and traveling waves are observed depending on the initial conditions and the payoff values.

The global phase portrait of structurally stable two-cell cellular neural networks is studied. The configuration of equilibrium points, the number of limit cycles and their locations are investigated systematically.

We consider a Cellular Neural Network (CNN), with a bias term, on the integer lattice ℤ² in the plane ℝ². Space-dependent, asymmetric couplings (templates) appropriate for CNN in the hexagonal lattice on ℝ² are studied. We characterize the mosaic patterns and study their spatial entropy. It appears that for this problem, asymmetry of the template has a more robust effect on the spatial entropy than does the sign of a parameter in the templates.

This presentation investigates the dynamics of discrete-time cellular neural networks (DT-CNN). In contrast to classical neural networks that are mostly gradient-like systems, DT-CNN possesses both complete stability and chaotic behaviors as different parameters are considered. An energy-like function which decreases along orbits of DT-CNN as well as the existence of a globally attracting set are derived. Complete stability can then be concluded, with further analysis on the sets on which the energy function is constant. The formations of saturated stationary patterns for DT-CNN are shown to be analogous to the ones in continuous-time CNN. Thus, DT-CNN shares similar properties with continuous-time CNN. By confirming the existence of snap-back repellers, hence transversal homoclinic orbits, we also conclude that DT-CNN with certain parameters exhibits chaotic dynamics, according to the theorem by Marotto.

Coupled map lattices with multidimensional lattice are considered. A method for the determination of the stability of spatially homogeneous and spatially periodic steady-state solutions is derived. This method is based on the determination of the spectrum of the linearized operator by means of Gelfand transformation of some appropriate Banach algebra. The results are applied to several examples.

We investigated the dynamical characteristics of neuromagnetic responses by recording magnetoencephalographic (MEG) signals to equiluminant flickering stimulus of different color combinations from a group of control subjects, and from a patient with photosensitive epilepsy. By wavelet based time-frequency analysis, we showed that two distinct neuromagentic responses corresponding to stimulus frequency and its time delayed first harmonic were found in control subjects, whereas no harmonic response was obtained for the patient. We applied a battery of methods (sample entropy measuring signal complexity and index of smoothness measuring determinism) based on nonlinear dynamical system theory in conjunction with bootstrapping surrogate analysis. The results suggested that a significant nonlinear structure was evident in the MEG signals for control subjects, whereas nonlinearity was not detected for the patient. In addition, the couplings between distant cortical regions were found to be greater for control subjects. The important role of combinational chromatic sensitivity in sustained cortical excitation was also confirmed. These findings lead to the hypothesis that the healthy human brain is most likely equipped with significantly nonlinear neuronal processing reflecting an inherent mechanism defending against hyper-excitation to chromatic flickering stimulus, and such nonlinear mechanism is likely to be impaired for a patient with photosensitive epilepsy.

In this paper, a methodology to design a system that robustly synchronizes a master chaotic system from a sampled driving signal is developed. The method is based on the fuzzy Takagi–Sugeno representation of chaotic systems, from which a continuous-time fuzzy observer is designed as the solution of an LMI minimization problem such that the error dynamics have H_∞ disturbance attenuation performance. Then, from the dual-system approach, the fuzzy observer is digitally redesigned such that the performance is maintained for the sampled master system. The effectiveness of the proposed synchronization methodology is finally illustrated via numerical simulations of the chaotic Chen's system.

In this paper, coupling properties of regular and chaotic calcium oscillations are examined. Synchronized calcium signals among coupled cells in tissue, where calcium ions were found to be one of the most important second messengers, have proven indispensable for proper and reliable functioning of living organisms. When modeling such systems, it is of particular interest to determine, which internal system properties guarantee best coupling abilities and herewith physiologically the most efficient signal transduction between cells. We found that local contractive properties of attractors in phase space, quantified by the local divergence, represent one of the crucial system properties that determine synchronization abilities of coupled regular and chaotic oscillators. In particular, parts of attractors with close to zero local divergence largely facilitate synchronization of initially unsynchronized oscillators. For bursting oscillations, this is fully in agreement with previous studies showing that synchronization abilities of bursters are closely related with the slow passage effect. We extended this concept with the help of local divergence and succeeded to apply our theory also to other oscillatory regimes, like regular spiking and complex chaotic oscillations.

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

This paper investigates the bifurcations due to time delay in the feedback control system with excitation. Based on an self-sustained oscillator, the delayed velocity feedback control system is proposed. For the case without excitation, the stability of the trivial equilibrium is discussed and the condition under which the equilibrium loses its stability is obtained. This leads to a critical stability boundary where Hopf bifurcation or periodic solutions may occur. For the case with excitation, the main attention is focused on the effect of time delay on the obtained periodic solution when primary resonance occurs in the system under consideration. To this end, the control system is changed to be a functional differential equation. Functional analysis is carried out to obtain the center manifold and then a perturbation approach is used to find periodic solutions in a closed form. Moreover, the unstable regions for the limit cycles are also obtained, predicting the occurrence of some complex behaviors. Numerical simulations are employed to find the routes leading to quasi-periodic motions as the time delay is varied. It has been found that: (i) Time delay can be used to control bifurcations; and (ii) time delay can be applied to generate bifurcations. This indicates that time delay may be used as a "switch" to control or create complexity for different applications.

Linear stability and Hopf bifurcation in a two-neuron network with three delays (one is due to the self-feedback) are investigated. Based on the normal form approach and the center manifold theory, we derive the formula to determine the direction and stability of Hopf bifurcation.

This paper presents a study of the behavior of a special class of 3D dynamic systems (i.e. RHS of the third-order equation is a cubic polynomial), using Lyapunov–Andronov's theory. Considering the general case, we find a new analytical formula for the first Lyapunov's value at the boundary of stability. It enables one to study in detail the bifurcation behavior (and the route to chaos, in particular) of dynamic systems of the above type. We check the validity of our analytical results on the first Lyapunov's value by studying the route to chaos of two 3D dynamic systems with proved chaotic behavior. These are Chua's and Rucklidge's systems. Considering their route to chaos, we find new results.

This paper investigates periodic bifurcation solutions of a mechanical system which involves a van der Pol type damping and a hysteretic damper representing restoring force. This system has recently been studied based on the singularity theory for bifurcations of smooth functions. However, the results do not actually take into account the property of nonsmoothness involved in the system. In particular, the transition varieties due to constraint boundaries were ignored, resulting in failure in finding some important bifurcation solutions. To reveal all possible bifurcation patterns for such systems, a new method is developed in this paper. With this method, a continuous, piecewise smooth bifurcation problem can be transformed into several subbifurcation problems with either single-sided or double-sided constraints. Further, the constrained bifurcation problems are converted to unconstrained problems and then singularity theory is employed to find transition varieties. Explicit formulas are applied to reconsider the mechanical system. Numerical simulations are carried out to verify analytical predictions. Moreover, symbolic notation for a sequence of bifurcations is introduced to easily show the characteristics of bifurcations, and also the comparison of different bifurcations. The method developed in this paper can be easily extended to study bifurcation problems with other types of nonsmoothness.

In this paper we present a relationship between the algebraic notion of proper system, the geometric notion of contact point and the dynamic notion of Poincaré map for piecewise linear differential systems. This allows to present sufficient conditions (which are also necessary under additional hypotheses) for the existence of Poincaré maps in piecewise linear differential systems. Moreover, an adequate parametrization of the Poincaré maps make such maps invariant under linear transformations.

The dynamics of differential-difference-algebraic equations is studied. The paper extends the study of the local bifurcations to transcritical and pitchfork bifurcation under certain nondegenerate conditions using Lyapunov–Schmidt reduction. Furthermore, an improved version of singularity induced bifurcation theorem is given in this paper.

We explore the chaotic behavior of a nonlinear electrical circuit constructed with simple components such as diodes and linear operational amplifiers. The circuit may be regarded as a nonlinear analog computer that gives a nearly exact solution of a particular chaotic model. Detailed comparisons between theoretical and experimental bifurcation points and power spectra yield differences of less than 1%.

How fluctuations can be eliminated or attenuated is a matter of general interest in the study of steadily-forced dissipative nonlinear dynamical systems. Here, we extend previous work on "nonlinear quenching" [Hide, 1997] by investigating the phenomenon in systems governed by the novel autonomous set of nonlinear ordinary differential equations (ODE's) , ẏ=-xzq+bx-y and ż=xyq-cz (where (x, y, z) are time(t)-dependent dimensionless variables and , etc.) in representative cases when q, the "quenching function", satisfies q=1-e+ey with 0≤e≤1. Control parameter space based on a,b and c can be divided into two "regions", an S-region where the persistent solutions that remain after initial transients have died away are steady, and an F-region where persistent solutions fluctuate indefinitely. The "Hopf boundary" between the two regions is located where b=b_H(a, c; e) (say), with the much studied point (a, b, c)=(10, 28, 8/3), where the persistent "Lorenzian" chaos that arises in the case when e=0 was first found lying close to b=b_H(a, c; 0). As e increases from zero the S-region expands in total "volume" at the expense of F-region, which disappears altogether when e=1 leaving persistent solutions that are steady throughout the entire parameter space.

In a recent paper Moroz [2003] considered a simplified version of third class self-exciting Faraday-disk dynamo model, introduced by Hide [1997], in the limit in which leads to the Malkus–Robbins dynamo [Malkus, 1972; Robbins, 1977] as a special case. In that study a linear series motor was incorporated which led to an enriching range of possible behavior that the original Malkus–Robbins dynamo could support. In this paper, we replace the linear motor by a nonlinear motor and consider the consequences on the dynamics of the dynamo.

In many models of pest control, increases in pest population due to birth are assumed to be continuous, but in fact, pest population reproduces only during a single period; at the same time, pesticides are often applied during the period. So in this paper we propose a ratio-dependent predator–prey model with birth pulse and pesticide pulse. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution of systems which have Ricker functions or Beverton–Holt functions, and obtain the threshold conditions for their stability. Above the threshold, there is a characteristic sequence of bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the ratio-dependent predator–prey model with birth pulse and pesticide pulse are very complex, including small-amplitude oscillations, large-amplitude cycles and chaos. This suggests that birth pulse and pesticide pulse, in effect, provide a natural period or cyclicity that allows for period-doubling bifurcation and period-halving bifurcation route to chaos.

The Hopf bifurcation, saddle connection loop bifurcation and Poincaré bifurcation of the generalized Rayleigh–Liénard oscillator Ẍ+aX+2bX³+ε(c₃+c₂X²+c₁X⁴+c₄Ẋ²)Ẋ=0 are studied. It is proved that for the case a<0, b>0 the system has at most six limit cycles bifurcated from Hopf bifurcation or has at least seven limit cycles bifurcated from the double homoclinic loop. For the case a>0, b<0 the system has at most three limit cycles bifurcated from Hopf bifurcation or has three limit cycles bifurcated from the heteroclinic loop.

We show that a discrete dynamical system with delayed arguments that describe the computational performance of a neural network with a loop structure can exhibit the coexistence of huge number of periodic solutions, and we describe the domains of attraction for the stable periodic orbits. This demonstrates a great potential of discrete time delayed neural nets with a loop structure for the purpose of storage and retrieval of periodic patterns.

In this paper, several simple and new conditions are presented to ensure existence, uniqueness and global exponential robust stability of the equilibrium point of neural networks with delays by using contraction mapping principle, homeomorphism property and Lyapunov method. The conditions given in this paper are easy to verify and apply, and the obtained results improve and extend to those in the literature. In addition, two examples are given to show the effectiveness of the proposed results.

In this letter, the characteristics of a novel switched circuit is presented. Its pseudo chaotic signal is digitally programmable. Robust synchronization between two mismatched circuits is demonstrated by simulation results. Lastly, applications of this circuit on chaotic communication are suggested.

We consider a model-based approach for the anticontrol of some discrete-time systems. We first assume the existence of a chaotic model in an appropriate form. Then by using an appropriate control input we try to match the controlled system with the chaotic system model. We also give a procedure to generate the model chaotic systems in arbitrary dimensions. We show that with this approach, controllable systems can always be chaotified. Moreover, if the system to be controlled is stable, control input can be chosen arbitrarily small.

This article extends the range of applications of a previous paper [Leung & Liu, 2003]. Some new methods to suppress chaos are proposed so that the Melnikov function is modified only slightly for the easy elimination of simple zeros. The reasonability of the methods is analyzed by the previously established criteria. Some examples are also given.

In this paper, a method that combines impulsive control and sporadic coupling for the synchronization of two Lorenz systems is proposed. The time required to synchronize the Lorenz systems can be readily computed. With this scheme, the Lorenz systems can be synchronized quickly and the time interval between successive impulsive signals can be made larger. This will improve the efficiency of channel bandwidth utilization when applied to secure communications. The results obtained in this paper are very useful in the applications of chaos-based secure communication systems.

In the process of designing and constructing switching power converters, chaotic operations are often observed intermittently between long periods of regular operations. In practice, such intermittent chaotic operations can be eliminated by incorporating appropriate design measures to combat interference of spurious signals. In this paper, we explain the mechanism that causes "intermittent" chaos in a popular type of switching converters, namely, current-mode controlled switching converters. The circuit model used to study the phenomenon incorporates a coupling process through which a spurious signal is coupled to the current sensing and ramp compensation circuitry, resulting in a modulation of the compensation slope which causes the system to become unstable intermittently. We show that coupling of spurious signals into the compensation ramp can cause intermittent chaotic or subharmonic operations.

A concept of higher order complexity is proposed in this letter. If a randomness-finding complexity [Rapp & Schmah, 2000] is taken as the complexity measure, the first-order complexity is suggested to be a measure of randomness of the original time series, while the second-order complexity is a measure of its degree of nonstationarity. A different order is associated with each different aspect of complexity. Using logistic mapping repeatedly, some quasi-stationary time series were constructed, the nonstationarity degree of which could be expected theoretically. The estimation of the second-order complexity of these time series shows that the second-order complexities do reflect the degree of nonstationarity and thus can be considered as its indicator. It is also shown that the second-order complexities of the EEG signals from subjects doing mental arithmetic are significantly higher than those from subjects in deep sleep or resting with eyes closed.

Reconstruction of processes and fields from noisy data is to solve a set of linear algebraic equations. Three factors affect the accuracy of reconstruction: (a) a large condition number of the coefficient matrix, (b) high noise-to-signal ratio in the source term, and (c) no a priori knowledge of noise statistics. To improve reconstruction accuracy, the set of linear algebraic equations is transformed into a new set with minimum condition number and noise-to-signal ratio using the rotation matrix. The procedure does not require any knowledge of low-order statistics of noises. Several examples including highly distorted Lorenz attractor illustrate the benefit of using this procedure.