Dynamical properties of a particle in a wave packet: Scaling invariance and boundary crisis

Some dynamical properties present in a problem concerning the acceleration of particles in a wave packet are studied. The dynamics of the model is described in terms of a two-dimensional area preserving map. We show that the phase space is mixed in the sense that there are regular and chaotic regions coexisting. We use a connection with the standard map in order to find the position of the first invariant spanning curve which borders the chaotic sea. We find that the position of the first invariant spanning curve increases as a power of the control parameter with the exponent 2/3. The standard deviation of the kinetic energy of an ensemble of initial conditions obeys a power law as a function of time, and saturates after some crossover. Scaling formalism is used in order to characterise the chaotic region close to the transition from integrability to nonintegrability and a relationship between the power law exponents is derived. The formalism can be applied in many different systems with mixed phase space. Then, dissipation is introduced into the model and therefore the property of area preservation is broken, and consequently attractors are observed. We show that after a small change of the dissipation, the chaotic attractor as well as its basin of attraction are destroyed, thus leading the system to experience a boundary crisis. The transient after the crisis follows a power law with exponent −2.     

Transient times,resonances and drifts of attractors in dissipative rotational dynamics

In a dissipative system the time to reach an attractor is often influenced by the peculiarities of the model and in particular by the strength of the dissipation. As a dissipative model we consider the spin–orbit problem providing the dynamics of a triaxial satellite orbiting around a central planet and affected by tidal torques. The model is ruled by the oblateness parameter of the satellite, the orbital eccentricity, the dissipative parameter and the drift term. We devise a method which provides a reliable indication on the transient time which is needed to reach an attractor in the spin–orbit model; the method is based on an analytical result, precisely a suitable normal form construction. This method provides also information about the frequency of motion. A variant of such normal form used to parameterize invariant attractors provides a specific formula for the drift parameter, which in turn yields a constraint – which might be of interest in astronomical problems – between the oblateness of the satellite and its orbital eccentricity.     

The dimension of the global attractor for dissipative reaction-diffusion systems

Our aim in this note is to construct an exponential attractor of optimal (with respect to the dissipation parameter) fractal dimension for dissipative reaction-diffusion systems without conditions on the growth of the nonlinear term.     

Global attractors for von Karman evolutions with a nonlinear boundary dissipation

Dynamic von Karman equations with a nonlinear boundary dissipation are considered. Questions related to long time behaviour, existence and structure of global attractors are studied. It is shown that a nonlinear boundary dissipation with a large damping parameter leads to an existence of global (compact) attractor for all weak (finite energy) solutions. This result has been known in the case of full interior dissipation, but it is new in the case when the boundary damping is the main dissipative mechanism in the system. In addition, we prove that fractal dimension of the attractor is finite. The proofs depend critically on the infinite speed of propagation associated with the von Karman model considered.     

Gevrey regularity for a class of dissipative equations with applications to decay

In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.     

Hausdorff dimension of a random invariant set

The notion of random attractor for a dissipative stochastic dynamical system has recently been introduced. It generalizes the concept of global attractor in the deterministic theory. It has been shown that many stochastic dynamical systems associated to a dissipative partial differential equation perturbed by noise do possess a random attractor. In this paper, we prove that, as in the case of the deterministic attractor, the Hausdorff dimension of the random attractor can be estimated by using global Lyapunov exponents. The result is obtained under very natural assumptions. As an application, we consider a stochastic reaction-diffusion equation and show that its random attractor has finite Hausdorff dimension.     

A group theory derivation of the relativistic path integral and the “history-string” dynamics

We derive the Feynman path integral for relativistic elementary particles using group theory considerations. We apply an approach in which choosing a symmetry group (or semigroup) allows deriving the kinematics and dynamics of a particle including the state space and the propagator from it. The quantum properties of a particle appear from intertwining two representations of the symmetry (semi)group, one of which describes local properties of the particle and the other describes the particle as a whole. The path-integrallike dynamics appears when the symmetry semigroup has a structure similar to that of the relativistic analogue of the Galilei group (in which the Lorentz-invariant “proper time” plays the role of time) with translations replaced with the semigroup of trajectories (parameterized paths). The classical action in the weight functional of the path integral is defined by this semigroup up to couplings to gauge and/or gravitational fields. The obtained formalism is suitable for describing not only pointlike particles but also nonlocal objects of the “history-string” type, which, as previously shown, allow explaining quark confinement.     

Spiral chaos in the nonholonomic model of a Chaplygin top

This paper presents a numerical study of the chaotic dynamics of a dynamically asymmetric unbalanced ball (Chaplygin top) rolling on a plane. It is well known that the dynamics of such a system reduces to the investigation of a three-dimensional map, which in the general case has no smooth invariant measure. It is shown that homoclinic strange attractors of discrete spiral type (discrete Shilnikov type attractors) arise in this model for certain parameters. From the viewpoint of physical motions, the trace of the contact point of a Chaplygin top on a plane is studied for the case where the phase trajectory sweeps out a discrete spiral attractor. Using the analysis of the trajectory of this trace, a conclusion is drawn about the influence of “strangeness” of the attractor on the motion pattern of the top.     

Micro-chaos in digital control

Summary In this paper we analyze a model for the effect of digital control on one-dimensional, linearly unstable dynamical systems. Our goal is to explain the existence of small, irregular oscillations that are frequently observed near the desired equilibrium. We derive a one-dimensional map that captures exactly the dynamics of the continuous system. Using thismicro-chaos map, we prove the existence of a hyperbolic strange attractor for a large set of parameter values. We also construct an “instability chart” on the parameter plane to describe how the size and structure of the chaotic attractor changes as the parameters are varied. The applications of our results include the stick-and-slip motion of machine tools and other mechanical problems with locally negative dissipation.     

Averaging the trajectory attractor of a nonlinear wave equation with rapidly oscillating right-hand side

We consider a nonlinear nonautonomous hyperbolic equation with dissipation and with a small parameter multiplying the highest derivative with respect to time. This equation also involves a rapidly oscillating external force. Using a standard technique for constructing the trajectory attractor, we can prove the convergence of the attractor of a nonlinear nonautonomous hyperbolic equation with dissipation to the attractor of the corresponding parabolic equation.     

Irreversible thermodynamics,quantum mechanics and intrinsic time scales

It is often assumed that phenomenology is a rather weak tool for the analysis of natural systems because it lacks generality. However, in a series of papers we have developed a phenomenological calculus based upon a general theory of measurement and mathematical representations (or, equivalently, upon system response as a bilinear form) which has a broad range of application. The present paper illustrates its power and versatility by demonstrating that irreversible thermodynamics and quantum mechanics are homomorphic. This result is, in itself, interesting since it shows that a large class of dissipative, deterministic systems are homomorphic to a large class of ideal, stochastic systems. In both cases, the metrical structure of the phenomenological calculus allows us to define a “proper time” intrinsic to the system dynamics. With this intrinsic time, a dynamics of aging can be defined upon the system's parameter space. In this context, Schrödinger's equation is seen as a dynamics of aging.     

Modeling of local particle accumulation in disperse flows with kinematic singularities

The aim of the study is to model the formation of local particle accumulation zones near several typical kinematic singularities. The flows considered are: (i) a steady two–dimensional flow with localized vorticity of the Kelvin cat's eye type (vortex in a mixing layer), (ii) a steady axisymmetric flow formed by a vortex filament normal to a plane in viscous fluid (simple model of tornado), (iii) a neighbourhood of a zero acceleration point in two–dimensional unsteady (harmonic) flow. From parametric numerical calculations, we investigated the inertial mechanisms of forming local particle accumulation zones and found the threshold values of governing parameters separating qualitatively different particle velocity and density patterns. In particular, it is shown that the zero–acceleration point can either “attract” or “scatter” the particles. Zones of concentrated vorticity are typically devoid of particles. In the tornado–like flow, an axisymmetric “cup-shaped” particle accumulation region is formed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A chaotic attractor in ecology: theory and experimental data

Chaos has now been documented in a laboratory population. In controlled laboratory experiments, cultures of flour beetles (Tribolium castaneum) undergo bifurcations in their dynamics as demographic parameters are manipulated. These bifurcations, including a specific route to chaos, are predicted by a well-validated deterministic model called the “LPA model”. The LPA model is based on the nonlinear interactions among the life cycle stages of the beetle (larva, pupa and adult). A stochastic version of the model accounts for the deviations of data from the deterministic model and provides the means for parameterization and rigorous statistical validation. The chaotic attractor of the deterministic LPA model and the stationary distribution of the stochastic LPA model describe the experimental data in phase space with striking accuracy. In addition, model-predicted temporal patterns on the attractor are observed in the data. This paper gives a brief account of the interdisciplinary effort that obtained these results.     

Global dynamics of nonautonomous Hindmarsh–Rose equations

Global dynamics of nonautonomous diffusive Hindmarsh–Rose equations on a three-dimensional bounded domain in neurodynamics is investigated. The existence of a pullback attractor is proved through uniform estimates showing the pullback dissipative property and the pullback asymptotical compactness. Then the existence of pullback exponential attractor is also established by proving the smoothing Lipschitz continuity in a long run of the solution process.     

Global dynamics of diffusive Hindmarsh–Rose equations with memristors

Global dynamics of the diffusive Hindmarsh–Rose equations with memristors as a new proposed model for neuron dynamics are investigated in this paper. We prove the existence and regularity of a global attractor for the solution semiflow through uniform analytic estimates showing the higher-order dissipative property and the asymptotically compact characteristics of the solutions by the approach of Kolmogorov–Riesz theorem. The quantitative bounds of the regions containing this global attractor respectively in the state space and in the regular space are explicitly expressed by the model parameters.     

（E0，E）型渐近光滑映射和耗散波方程

本文在无穷维空间引入（Ｅ_０，Ｅ）型渐近光滑映射的概念，研究了其基本性质和变为Ｅ中渐近光滑映射的条件，我们证明了（Ｅ_０，Ｅ）型吸引子存在性定理和（Ｅ_０，Ｅ）型吸引子转化为Ｅ中吸引子的条件定理，所有结果都应用于一类耗散波方程渐近性态的研究。映射，吸引子，耗散波     

A study of prebiotic evolution

A very simple model prebiotic system is explored, both, to elucidate the full complexity of its dynamic behavior, and from the standpoint of what a thermal analysis can tell us about evolution more generally. The system consists of a coacervate containing a four variable enzymatic oscillator that is driven by a single input concentration. The reaction scheme is “nested”; i.e., by “turning on” one reaction at a time we can go from a two‐variable system, to one of three variables, to one of four, each developing more complex behavior. The four‐variable system is shown to have at least five distinct genera of complex attractors within the range examined. Two of these coexist for the same parameter values; the other three are substantially separated in the bifurcation space. The fixed points, characteristic roots, Lyapunov exponents, stability (dissipation), Kaplan‐Yorke dimensions, and correlation dimensions are all calculated for each attractor. A six‐variable amplification of the reaction scheme is considered as a simple model of nucleation in a coacervate and is shown to totally stabilize the corresponding attractor. An Evolutionary Potential is proposed that is wholly beyond the purview of classical thermostatics, yet incorporates entropy effects via Clausius' strong version of the Second Law. It is shown that the latter is a necessary condition for the sort of structuring characteristic of living systems. © 2003 Wiley Periodicals, Inc. Complexity 8: 45–67, 2003     

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L ²-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.     

Dynamical phenomena occurring due to phase volume compression in nonholonomic model of the rattleback

We study numerically the dynamics of the rattleback, a rigid body with a convex surface on a rough horizontal plane, in dependence on the parameters, applying methods used earlier for treatment of dissipative dynamical systems, and adapted here for the nonholonomic model. Charts of dynamical regimes on the parameter plane of the total mechanical energy and the angle between the geometric and dynamic principal axes of the rigid body are presented. Characteristic structures in the parameter space, previously observed only for dissipative systems, are revealed. A method for calculating the full spectrum of Lyapunov exponents is developed and implemented. Analysis of the Lyapunov exponents of the nonholonomic model reveals two classes of chaotic regimes. For the model reduced to a 3D map, the first one corresponds to a strange attractor with one positive and two negative Lyapunov exponents, and the second to the chaotic dynamics of quasi-conservative type, when positive and negative Lyapunov exponents are close in magnitude, and the remaining exponent is close to zero. The transition to chaos through a sequence of period-doubling bifurcations relating to the Feigenbaum universality class is illustrated. Several examples of strange attractors are considered in detail. In particular, phase portraits as well as the Lyapunov exponents, the Fourier spectra, and fractal dimensions are presented.     

Dissipative relativistic standard map: Periodic attractors and basins of attraction

The dissipative relativistic standard map, introduced by Ciubotariu et al. [Ciubotariu C, Badelita L, Stancu V. Chaos in dissipative relativistic standard maps. Chaos, Solitons & Fractals 2002;13:1253–67.], is further studied numerically for small damping in the resonant case. We find that the attractors are all periodic; their basins of attraction have fractal boundaries and are closely interwoven. The number of attractors increases with decreasing damping. For a very small damping, there are thousands of periodic attractors, comprising mostly of the lowest-period attractors of period one or two; the basin of attraction of these lowest-period attractors is significantly larger compared to the basins of the higher-period attractors.