Singularity Structure and Chaotic Behavior of the Homopolar Disk Dynamo

We study in great detail a system of three first-order ordinary differential equations describing a homopolar disk dynamo (HDD). This system displays a large variety of behaviors, both regular and chaotic. Existence of periodic solutions is proved for certain ranges of parameters. Stability criteria for periodic solutions are given. The nonintegrability aspects of the HDD system are studied by investigating analytically the singularity structure of the system in the complex domain. Coexisting attractors (including period-doubling sequence) and coexisting strange attractors appear in some parametric regimes. The gluing of strange attractors and the ungluing of a strange attractor are also shown to occur. A period of bifurcation leading to chaos, not observed for other chaotic systems, is shown to characterize the chaotic behavior in some parametric ranges. The limiting case of the Lorenz system is also studied and is related to HDD.     

Multiple attractors and crisis route to chaos in a model food-chain

An attempt has been made to identify the mechanism, which is responsible for the existence of chaos in narrow parameter range in a realistic ecological model food-chain. Analytical and numerical studies of a three species food-chain model similar to a situation likely to be seen in terrestrial ecosystems has been carried out. The study of the model food chain suggests that the existence of chaos in narrow parameter ranges is caused by the crisis-induced sudden death of chaotic attractors. Varying one of the critical parameters in its range while keeping all the others constant, one can monitor the changes in the dynamical behaviour of the system, thereby fixing the regimes in which the system exhibits chaotic dynamics. The computed bifurcation diagrams and basin boundary calculations indicate that crisis is the underlying factor which generates chaotic dynamics in this model food-chain. We investigate sudden qualitative changes in chaotic dynamical behaviour, which occur at a parameter value a₁=1.7804 at which the chaotic attractor destroyed by boundary crisis with an unstable periodic orbit created by the saddle-node bifurcation. Multiple attractors with riddled basins and fractal boundaries are also observed. If ecological systems of interacting species do indeed exhibit multiple attractors etc., the long term dynamics of such systems may undergo vast qualitative changes following epidemics or environmental catastrophes due to the system being pushed into the basin of a new attractor by the perturbation. Coupled with stochasticity, such complex behaviours may render such systems practically unpredictable.     

一类平面D_3等变映射的混沌吸引子对称增加分歧的数值确定

本文讨论了一类平面D3等变映射的分歧和混沌性质.通过计算显示出映射随着参数的变化,从周期解走向混沌以及混饨吸引子由Z2-对称走向D3-对称的全过程.给出计算混沌吸引子的对称增加分歧扩张系统的算法,数值结果表明,两者相符.     

Dynamics of Perturbed Wavetrain Solutions to the Ginzburg-Landau Equation

The bifurcation structure and asymptotic dynamics of even, spatially periodic solutions to the time-dependent Ginzburg-Landau equation are investigated analytically and numerically. All solutions spring from unstable periodic modulations of a uniform wavetrain. Asymptotic states include limit cycles, two-tori, and chaotic attractors. Lyapunov exponents for some chaotic motions are obtained. These show the solution strange attractors to have a fractal dimension slightly greater than 3.     

Occurrence and underlying mechanism of multi-stripe chaotic attractors

This paper is concerned with the generation of multi-stripe chaotic attractors. Simple periodic nonlinear functions are employed to transform the original chaotic attractors to a pattern with multiple “parallel” or “rectangular” stripes. The relationship between the system parameters related to some periodic functions and the shape of the generated attractor is analyzed. Theoretic analysis about the underlying mechanism of generating the parallel stripes in the attractors is given. A general creation mechanism of multi-stripe attractors of the Lorenz system and other well-known chaotic systems is derived from the proposed unified approach.     

Strange attractors in saddle-node cycles: prevalence and globality

 We consider parametrized families of diffeomorphisms bifurcating through the creation of critical saddle-node cycles. We show that they always exhibit Hénon-like strange attractors for a set of parameter values with positive Lebesgue density at the bifurcation value. In open classes of such families the strange attractors are of global type: their basins contain an a priori defined neighbourhood of the cycle. Furthermore, the bifurcation parameter may also be a point of positive density of hyperbolic dynamics. Oblatum VIII-1993 & 23-II-1995     

Generations and mechanisms of multi-stripe chaotic attractors of fractional order dynamic system

In this paper, the generations of multi-stripe chaotic attractors of fractional order system are considered. The original fractional order chaotic attractors can be turned into a pattern with multiple “parallel” or “ rectangular” stripes by employing certain simple periodic nonlinear functions. The relationships between the parameters relate to the periodic functions and the shape of the generated attractors are analyzed. Theoretical investigations about the underlying mechanisms of the parallel striped attractors of fractional order system are presented, with the fractional order Lorenz, Rössler and Chua’s systems as examples. Moreover, the periodic doubling striped route to chaos of fractional order Rössler system and maximum Lyaponov exponent calculations are also given.     

A four-wing chaotic attractor generated from a new 3-D quadratic autonomous system

This paper introduces a new 3-D quadratic autonomous system, which can generate two coexisting single-wing chaotic attractors and a pair of diagonal double-wing chaotic attractors. More importantly, the system can generate a four-wing chaotic attractor with very complicated topological structures over a large range of parameters. Some basic dynamical behaviors and the compound structure of the new 3-D system are investigated. Detailed bifurcation analysis illustrates the evolution processes of the system among two coexisting sinks, two coexisting periodic orbits, two coexisting single-wing chaotic attractors, major and minor diagonal double-wing chaotic attractors, and a four-wing chaotic attractor. Poincaré-map analysis shows that the system has extremely rich dynamics. The physical existence of the four-wing chaotic attractor is verified by an electronic circuit. Finally, spectral analysis shows that the system has an extremely broad frequency bandwidth, which is very desirable for engineering applications such as secure communications.     

Complex economic dynamics: Chaotic saddle, crisis and intermittency

Complex economic dynamics is studied by a forced oscillator model of business cycles. The technique of numerical modeling is applied to characterize the fundamental properties of complex economic systems which exhibit multiscale and multistability behaviors, as well as coexistence of order and chaos. In particular, we focus on the dynamics and structure of unstable periodic orbits and chaotic saddles within a periodic window of the bifurcation diagram, at the onset of a saddle-node bifurcation and of an attractor merging crisis, and in the chaotic regions associated with type-I intermittency and crisis-induced intermittency, in non-linear economic cycles. Inside a periodic window, chaotic saddles are responsible for the transient motion preceding convergence to a periodic or a chaotic attractor. The links between chaotic saddles, crisis and intermittency in complex economic dynamics are discussed. We show that a chaotic attractor is composed of chaotic saddles and unstable periodic orbits located in the gap regions of chaotic saddles. Non-linear modeling of economic chaotic saddle, crisis and intermittency can improve our understanding of the dynamics of financial intermittency observed in stock market and foreign exchange market. Characterization of the complex dynamics of economic systems is a powerful tool for pattern recognition and forecasting of business and financial cycles, as well as for optimization of management strategy and decision technology.     

Chaos, complex transients and noise: Illustration with a Kaldor model

In the present paper two-dimensional discrete Kaldor-type models are investigated. First, a sufficient condition for the existence of topological chaos of the model is derived analytically for a special parameter set. Second, the influences of noise on the Kaldor model are examined numerically. We show that noise may not only obscure the underlying structures, but also reveal the hidden structures, for example, the chaotic attractors near a window of chaos or the periodic attractors near a small chaotic parameter region.     

Replicate periodic windows in the parameter space of driven oscillators

In the bi-dimensional parameter space of driven oscillators, shrimp-shaped periodic windows are immersed in chaotic regions. For two of these oscillators, namely, Duffing and Josephson junction, we show that a weak harmonic perturbation replicates these periodic windows giving rise to parameter regions correspondent to periodic orbits. The new windows are composed of parameters whose periodic orbits have the same periodicity and pattern of stable and unstable periodic orbits already existent for the unperturbed oscillator. Moreover, these unstable periodic orbits are embedded in chaotic attractors in phase space regions where the new stable orbits are identified. Thus, the observed periodic window replication is an effective oscillator control process, once chaotic orbits are replaced by regular ones.     

Quasiperiodicity,strange non-chaotic and chaotic attractors in a forced two degrees-of-freedom system

Strange non-chaotic, strange chaotic and quasiperiodic attractors are demonstrated to exist for a system of two non-linear coupled oscillators with almost periodic excitations. For same parameter values a transition from a strange non-chaotic to a quasiperiodic attractor is presented, whereas for other parameter values a shift from the strange chaotic attractor to a quasiperiodic one is found.     

The evolution of a novel four-dimensional autonomous system: Among 3-torus, limit cycle, 2-torus, chaos and hyperchaos

In this paper, a novel four-dimensional autonomous system in which each equation contains a quadratic cross-product term is constructed. It exhibits extremely rich dynamical behaviors, including 3-tori (triple tori), 2-tori (quasi-periodic), limit cycles (periodic), chaotic and hyperchaotic attractors. In particular, we observe 3-torus phenomena, which have been rarely reported in four-dimensional autonomous systems in previous work. With the parameter r varying in quite a wide range, the evolution process of the system begins from 3-tori, and after going through a series of periodic, quasi-periodic and chaotic attractors in so many different shapes coming into being alternately, it evolves into hyperchaos, finally it degenerates to periodic attractor. Moreover, when the system is hyperchaotic, its two positive Lyapunov exponents are much larger than those of the hyperchaotic systems already reported, especially the largest Lyapunov exponents. We also observe a chaotic attractor of a very special shape. The complex dynamical behaviors of the system are further investigated by means of Lyapunov exponents spectrum, bifurcation diagram and phase portraits.     

Crisis of interspike intervals in Hodgkin–Huxley model

The bifurcations of the chaotic attractor in a Hodgkin–Huxley (H–H) model under stimulation of periodic signal is presented in this work, where the frequency of signal is taken as the controlling parameter. The chaotic behavior is realized over a wide range of frequency and is visualized by using interspike intervals (ISIs). Many kinds of abrupt undergoing changes of the ISIs are observed in different frequency regions, such as boundary crisis, interior crisis and merging crisis displaying alternately along with the changes of external signal frequency. And there are logistic-like bifurcation behaviors, e.g., periodic windows and fractal structures in ISIs dynamics. The saddle-node bifurcations resulting in collapses of chaos to period-6 orbit in dynamics of ISIs are identified.     

Striped attractor generation and synchronization analysis for coupled Rössler systems

A model of networked chaotic Rössler systems with periodic couplings is discussed. New phenomena, including individual attractors in striped rectangular shapes and partial synchronization (or clustering), are shown for these locally coupled systems. Coupling-induced attractors with multiple stripes can be easily controlled by coupling parameters. Moreover, various interconnection topologies are also taken into consideration in the synchronization analysis, and dynamical behaviors of the coupled systems are illustrated by numerical results.     

Rate of afferent stimulus dependent synchronization and coding in coupled neurons system

Sequences of intervals between firing times (interspike interval (ISI)) from a pair of locus ceruleus (LC) neurons coupled by axon–dendrite synapse with stimulus of constant and chaos are investigated in this paper. We analyze how the dynamical properties of chaotic input determine those of the output ISI sequences, and assess how various strength of stimulus and coupling affects the input–output relationship. The attractors constructed from delay embeddings of ISIs and of chaotic input are compared from the points of view of geometry and nonlinear dynamics characteristics, i.e., Lyapunov exponent spectrum (LES), Kaplan–York fractal dimension (KY_D) and unstable periodic orbit (UPO). For the coupled LC neurons system investigated, with the moderate strength of stimulus and coupling, the synchronous oscillation of the two neurons is well preserved even if the external stimulus is chaotic; the similarity between these attractors is high only when the afferent stimulus strength is smaller and rate is lower. When these conditions are satisfied, the output two ISI sequences are reciprocally related to input signals, and their oscillation wave shape in time course can be derived from that of the input signals variation, furthermore, the similar input sequence of order of UPOs, distribution of LES and value of KY_D remain in attractors reconstructed from ISI sequences. But these phenomena will disappear in higher rate of stimulus activity or in changing of the strength of stimulus and coupling, for this situation, the ISIs shows bifurcate behavior. These results may be of vital importance for any kind of information processing based on the neurons and temporal coding.     

The dynamics of an epidemic model for pest control with impulsive effect

In pest control, there are only a few papers on mathematical models of the dynamics of microbial diseases. In this paper a model concerning biologically-based impulsive control strategy for pest control is formulated and analyzed. The paper shows that there exists a globally stable susceptible pest eradication periodic solution when the impulsive period is less than some critical value. Further, the conditions for the permanence of the system are given. In addition, there exists a unique positive periodic solution via bifurcation theory, which implies both the susceptible pest and the infective pest populations oscillate with a positive amplitude. In this case, the susceptible pest population is infected to the maximum extent while the infective pest population has little effect on the crops. When the unique positive periodic solution loses its stability, numerical simulation shows there is a characteristic sequence of bifurcations, leading to a chaotic dynamic, which implies that this model has more complex dynamics, including period-doubling bifurcation, chaos and strange attractors.     

Homoclinic Tangles, Phase Locking, and Chaos in a Two-Frequency Perturbation of Duffing's Equation

Summary. We study a two-frequency perturbation of Duffing's equation. When the perturbation is small, this system has a normally hyperbolic invariant torus which may be subjected to phase locking. Applying a version of Melnikov's method for multifrequency systems, we detect the occurrence of transverse intersection between the stable and unstable manifolds of the invariant torus. We show that if the invariant torus is not subjected to phase locking, then such a transverse intersection yields chaotic dynamics. When the invariant torus is subjected to phase locking, the situation is different. In this case, there exist two periodic orbits which are created in a saddle-node bifurcation. Using another version of Melnikov's method for slowly varying oscillators, we also give conditions under which the stable and unstable manifolds of the periodic orbits intersect transversely and hence chaotic dynamics may occur. Our results reveal that when the invariant torus is subjected to phase locking, chaotic dynamics resulting from transverse intersection between its stable and unstable manifolds may be interrupted. Received November 18, 1993; final revision received September 9, 1997; accepted October 27,1997     

Bifurcations and Chaos in Duffing Equation

The Duffing equation with even-odd asymmetrical nonlinear-restoring force and one external forcingis investigated.The conditions of existence of primary resonance,second-order,third-order subharmonics,m-order subharmonics and chaos are given by using the second-averaging method,the Melnikov method andbifurcation theory.Numerical simulations including bifurcation diagram,bifurcation surfaces and phase portraitsshow the consistence with the theoretical analysis.The numerical results also exhibit new dynamical behaviorsincluding onset of chaos,chaos suddenly disappearing to periodic orbit,cascades of inverse period-doublingbifurcations,period-doubling bifurcation,symmetry period-doubling bifurcations of period-3 orbit,symmetry-breaking of periodic orbits,interleaving occurrence of chaotic behaviors and period-one orbit,a great abundanceof periodic windows in transient chaotic regions with interior crises and boundary crisis and varied chaoticattractors.Our results show that many dynamical behaviors are strictly departure from the behaviors of theDuffing equation with odd-nonlinear restoring force.     

Mechanism of producing a saddle-node bifurcation with the coalescence of two unstable periodic orbits

A saddle-node bifurcation with the coalescence of a stable periodic orbit and an unstable periodic orbit is a common phenomenon in nonlinear systems. This study investigates the mechanism of producing another saddle-node bifurcation with the coalescence of two unstable periodic orbits. The saddle-node bifurcation results from a codimension-two bifurcation that a period doubling bifurcation line tangentially intersects a saddle-node bifurcation line in a parameter plane. Based on the bifurcation theory, the saddle-node bifurcation with the coalescence of two unstable periodic orbits is studied using the codimension-two bifurcation.