Intermittency in dissipative dynamical systems

We show that the intermittency displayed by a differential system proposed by Yamada and Fujisaka can be interpreted in the general framework of intermittent transitions to turbulence studied by Pomeau and the author.     

Bifurcation and chaos in simple jerk dynamical systems

In recent years, it is observed that the third-order explicit autonomous differential equation, named as jerk equation, represents an interesting sub-class of dynamical systems that can exhibit many major features of the regular and chaotic motion. In this paper, we investigate the global dynamics of a special family of jerk systems {ie075-01}, whereG(x) is a non-linear function, which are known to exhibit chaotic behaviour at some parameter values. We particularly identify the regions of parameter space with different asymptotic dynamics using some analytical methods as well as extensive Lyapunov spectra calculation in complete parameter space. We also investigate the effect of weakening as well as strengthening of the non-linearity in theG(x) function on the global dynamics of these jerk dynamical systems. As a result, we reach to an important conclusion for these jerk dynamical systems that a certain amount of non-linearity is sufficient for exhibiting chaotic behaviour but increasing the non-linearity does not lead to larger regions of parameter space exhibiting chaos.     

Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

Extensive and subextensive chaos in globally coupled dynamical systems

Using a combination of analytical and numerical techniques, we show that chaos in globally coupled identical dynamical systems, whether dissipative or Hamiltonian, is both extensive and subextensive: their spectrum of Lyapunov exponents is asymptotically flat (thus extensive) at the value λ(0) given by a single unit forced by the mean field, but sandwiched between subextensive bands containing typically O(logN) exponents whose values vary as λ?λ(∞)+c/logN with λ(∞)≠λ(0).     

Transition to chaos in continuous-time random dynamical systems

We consider situations where, in a continuous-time dynamical system, a nonchaotic attractor coexists with a nonattracting chaotic saddle, as in a periodic window. Under the influence of noise, chaos can arise. We investigate the fundamental dynamical mechanism responsible for the transition and obtain a general scaling law for the largest Lyapunov exponent. A striking finding is that the topology of the flow is fundamentally disturbed after the onset of noisy chaos, and we point out that such a disturbance is due to changes in the number of unstable eigendirections along a continuous trajectory under the influence of noise.     

Correlation properties of dynamical chaos in Hamiltonian systems

The structure of chaos border in phase space and its impact on the correlation and other statistical properties of the chaotic motion are considered. We conjecture that such a structure is described by a chaotic renormalization group. The effects of an external noise and of dissipation are discussed.     

Normal forms of reversible dynamical systems

We consider reversible dynamical systems with a fixed point which is also fixed under the reversing involution; we show that applying to such a system the canonical Poincaré-Dulac procedure reducing a dynamical system to its normal form, we obtain a normal form which is still reversible (under the same involution as the original system); conversely, we also show how to obtain all the reversible systems which are reduced to a given reversible form. This allows one to (locally) classify reversible dynamical systems, and reduce their (local) study to that of reversible normal forms.     

Complex dynamical behavior and chaos control in fractional-order Lorenz-like systems

In this paper, the complex dynamical behavior of a fractional-order Lorenz-like system with two quadratic terms is investigated. The existence and uniqueness of solutions for this system are proved, and the stabilities of the equilibrium points are analyzed as one of the system parameters changes. The pitchfork bifurcation is discussed for the first time, and the necessary conditions for the commensurate and incommensurate fractional-order systems to remain in chaos are derived. The largest Lyapunov exponents and phase portraits are given to check the existence of chaos. Finally, the sliding mode control law is provided to make the states of the Lorenz-like system asymptotically stable. Numerical simulation results show that the presented approach can effectively guide chaotic trajectories to the unstable equilibrium points.     

Interference and spontaneous emission

The field structure of the spontaneously emitted radiation from a single two-level atom is calculated in terms of atomic source variables and used to discuss the coherence properties of spontaneous emission.     

一种非线性动力学系统混沌现象的深入研究

本文在实验教学中引入一种非线性混沌摆系统,通过调节混沌摆的驱动力周期演示了该非线性动力学系统出现混沌现象的过程,从而让学生了解混沌现象的参数敏感性、相图特点、频谱特性等基本特性.为了进一步了解该混沌摆的特性,本文建立了该非线性摆系统的简化动力学方程,在数值上对其进行了研究.基于动力学方程的数值模拟,克服了实验上相关参数定量改变困难、摆动稳定性不易控制、实验时间周期长等问题.在数值模拟中,通过改变不同参数得到了相图、频谱图以及分岔图,比较深入详细地对这种混沌摆的相关特性进行了描述,也有利于学生加深对混沌摆的理解.     

Quantum fluctuations and dynamical chaos: An effective potential approach

We discuss the intimate connection between the chaotic dynamics of a classical field theory and the instability of the one-loop effective action of the associated quantum field theory. Using the example of massless scalar electrodynamics, we show how the radiatively induced spontaneous symmetry breaking stabilizes the vacuum state against chaos, and we speculate that monopole condensation can have the same effect in non-Abelian gauge theories.     

Using invariants to change detection in dynamical system with chaos

Change detection is the crucial subject in dynamical systems. There are suitable methods for detecting changes for linear systems and some methods for nonlinear systems, but there is a lack of methods concerning chaotic systems. This paper presents change detection techniques for dynamical systems with chaos. We consider the dynamical system described by the time series which originated from ordinary differential equation and real-world phenomena. We assume that the change parameters are unknown and the change could be either slight or drastic. The process of change detection is based on characteristic dynamical system invariants. Changes in the invariants’ values of the dynamical systems are the indicators of change. We propose a method of change detection based on the fractal dimension and recurrence plot. The automatic detection is provided by control charts. Methods were checked by using small data sets and stream data.     

Classical gauge theories as dynamical systems—Regularity and chaos

In this review we present the salient features of dynamical chaos in classical gauge theories with spatially homogeneous fields. The chaotic behaviour displayed by both abelian and non-abelian gauge theories and the effect of the Higgs term in both cases are discussed. The role of the Chern-Simons term in these theories is examined in detail. Whereas, in the abelian case, the pure Chern-Simons-Higgs system is integrable, addition of the Maxwell term renders the system chaotic. In contrast, the non-abelian Chern-Simons-Higgs system is chaotic both in the presence and the absence of the Yang-Mills term. We support our conclusions with numerical studies on plots of phase trajectories and Lyapunov exponents. Analytical tests of integrability such as the Painlevé criterion are carried out for these theories. The role of the various terms in the Hamiltonians for the abelian Higgs, Yang-Mills-Higgs and Yang-Mills-Chern-Simons-Higgs systems with spatially homogeneous fields, in determining the nature of order-disorder transitions is highlighted, and the effects are shown to be counter-intuitive in the last-named system.     

Classical dynamical simulation of spontaneous alloying

“Spontaneous alloying” observed by Yasuda, Mori et al. for metallic small clusters is simulated using classical Hamiltonian dynamics. Very rapid alloying occurs homogeneously and cooperatively starting from the solid phase of the cluster if the heat of solution is negative and the size of cluster is less than a critical size. Analysis of 2D models reveals that the alloying rate obeys an Arrhenius-type law, which predicts the alloying time much less than second at room temperature. Evidences manifesting that the spontaneous alloying proceeds in the solid phase without melting are also presented. The simulation reproduces the essential features of the experiments. Received: 2 March 1998 / Revised: 21 May 1998 / Accepted: 28 May 1998