准周期外力驱动下Lorenz系统的动力学行为

本文研究了准周期外力驱动下Lorenz系统的动力学行为，发现当外强迫的振幅达到某一个临界值时，系统的动力学行为将会发生根本性的变化，由此揭示了产生非混沌奇怪吸引子(Strange Nonchaotic Attractor, SNA)的一个新机制：准周期外强迫振幅的加大导致系统由奇怪的混沌吸引子转变为SNA，系统的相空间最终被压缩至一个准周期环上.并且本文的结果表明，外强迫的临界振幅与Lorenz系统Rayleigh数的大小成正比，而其受外强迫频率变化的影响并不大. 关键词： 准周期 Lorenz系统 非混沌奇怪吸引子     

Data-Driven Corrections of Partial Lotka–Volterra Models

In many applications of interacting systems, we are only interested in the dynamic behavior of a subset of all possible active species. For example, this is true in combustion models (many transient chemical species are not of interest in a given reaction) and in epidemiological models (only certain subpopulations are consequential). Thus, it is common to use greatly reduced or partial models in which only the interactions among the species of interest are known. In this work, we explore the use of an embedded, sparse, and data-driven discrepancy operator to augment these partial interaction models. Preliminary results show that the model error caused by severe reductions—e.g., elimination of hundreds of terms—can be captured with sparse operators, built with only a small fraction of that number. The operator is embedded within the differential equations of the model, which allows the action of the operator to be interpretable. Moreover, it is constrained by available physical information and calibrated over many scenarios. These qualities of the discrepancy model—interpretability, physical consistency, and robustness to different scenarios—are intended to support reliable predictions under extrapolative conditions.     

On computing the entropy of the Henon attractor

In a recent article D. Ruelle [inLecture Notes in Physics, No. 80 (Springer, Berlin, 1978)] has conjectured that for the Hénon attractor its measure theoretic entropy should be equal to its characteristic exponent. This result is known to be true for systems which satisfy Smale's Axiom A. In this article we report the results of our computations which suggest that Ruelle's conjecture may be true for the Hénon attractor. Further, in our study we are confronted with fundamental questions which suggest that certain existence theorems from ergodic theory are not sufficient from a computational point of view.     

Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.     

Sharp Lower Bounds for the Dimension of the Global Attractor of the Sabra Shell Model of Turbulence

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log  ν ⁻¹ for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity. AMS Subject Classifications: 76F20, 76D05, 35Q30     

Estimating the bound for the generalized Lorenz system

A chaotic system is bounded, and its trajectory is confined to a certain region which is called the chaotic attractor. No matter how unstable the interior of the system is, the trajectory never exceeds the chaotic attractor. In the present paper, the sphere bound of the generalized Lorenz system is given, based on the Lyapunov function and the Lagrange multiplier method. Furthermore, we show the actual parameters and perform numerical simulations.     

Detection of Mechanism of Noise-Induced Synchronization between Two Identical Uncoupled Neurons

We investigate the noise-induced synchronization between two identical uncoupled Hodgkin-Huxley neurons with sinusoidal stimulations. The numerical results confirm that the value of critical noise intensity for synchronizing two systems is much less than the magnitude of mean size of the attractor in the original system, and the deterministic feature of the attractor in the original system remains unchanged. This finding is significantly different from the previous work [Phys. Rev. E 67 （2003） 027201] in which the value of the critical noise intensity for synchronizing two systems was found to be roughly equal to the magnitude of mean size of the attractor in the original system, and at this intensity, the noise swamps the qualitative structure of the attractor in the original deterministic systems to synchronize to their stochastic dynamics. Further investigation shows that the critical noise intensity for synchronizing two neurons induced by noise may be related to the structure of interspike intervals of the original systems.     

Dissipation effect on chaos onset in a two-resonance overlap case

The problem of two-resonance interaction is considered in the dissipative case. A strange attractor is shown to appear under certain conditions. Hierarchy substructures are obtained when the strange attractor degenerates.     

Experimental versus theoretical robustness of rotating solutions in a parametrically excited pendulum: A dynamical integrity perspective

The objective of this paper is showing how global safety arguments can be fruitfully used to interpret experimental results of a pendulum parametrically excited by wave motion. In fact, the results of an experimental campaign developed with the aim of simulating sea-waves energy production by a parametric pendulum show that rotations exist in a region which is smaller than the theoretical one. This discrepancy can be partially attributed to the experimental approximations and constraints, but it has a deeper theoretical motivation. By comparing the experimental results with the dynamical integrity profiles we have found that experimental rotations exist only where a measure of dynamical integrity accounting for both attractor robustness and basin compactness is large enough, so that they can support experimental imperfections leading to changes in initial conditions.     

均匀湍流中的奇怪吸引子

本文对于实测的湍流信号采用延迟坐标重建相空间技术计算了它的关联维数,熵和最大Лялнов指数,从而指出这是一种受随机噪声干扰的确定性的混沌现象,它在相空间的吸引子是一个随机噪声背景上的奇怪吸引子。     

Distribution of particles and bubbles in turbulence at a small Stokes number

The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multifractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally.     

A study of hydrodynamic turbulence using laser transmission

Using laser transmission, the characteristics of hydrodynamic turbulence is studied following one of the recently developed technique in nonlinear dynamics. The existence of deterministic chaos in turbulence is proved by evaluating two invariants viz. dimension of attractor and Kolmogorov entropy. The behaviour of these invariants indicates that above a certain strength of turbulence the system tends to more ordered states.     

Multistable randomly switching oscillators: The odds of meeting a ghost

We consider oscillators whose parameters randomly switch between two values at equal time intervals. If random switching is fast compared to the oscillator’s intrinsic time scale, one expects the switching system to follow the averaged system, obtained by replacing the random variables with their mean. The averaged system is multistable and one of its attractors is not shared by the switching system and acts as a ghost attractor for the switching system. Starting from the attraction basin of the averaged system’s ghost attractor, the trajectory of the switching system can converge near the ghost attractor with high probability or may escape to another attractor with low probability. Applying our recent general results on convergent properties of randomly switching dynamical systems [1, 2], we derive explicit bounds that connect these probabilities, the switching frequency, and the chosen initial conditions.     

Fluctuation-induced escape from the basin of attraction of a quasiattractor

Noise-induced escape from the basin of attraction of a strange attractor (SA) in a periodically excited nonlinear oscillator is investigated. It is shown by numerical simulation methods that escape occurs in two steps: transfer of the system from the SA to a close-lying saddle cycle along several optimal trajectories, and a subsequent fluctuation-induced transfer from the basin of attraction of the SA along a single optimal trajectory. The possibility of using the results of this work to solve problems of the optimal control of switchings from an attractor and for constructing theoretical estimates of the escape probability is discussed. Pis’ma Zh. éksp. Teor. Fiz. 69, No. 11, 782–787 (10 June 1999)     

Study of a high-dimensional chaotic attractor

The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the exterior subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace goes out of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.     

一类可变禁区的不连续系统的加周期分岔

研究了一类可变禁区不连续系统的加周期分岔行为, 发现由可变禁区导致不同类型的加周期分岔. 研究表明, 系统的迭代轨道和禁区的上下两个边界均可发生边界碰撞, 从而产生加周期分岔. 基于边界碰撞分岔理论, 定义基本的迭代单元, 解析推导出了相应的分岔曲线, 在全参数空间中给出了不同加周期所出现的范围. 与数值模拟结果比较, 理论分析结果与数值结果高度一致.     

Comments on the Entropy of Nonequilibrium Steady States

We discuss the entropy of nonequilibrium steady states. We analyze the so-called spontaneous production of entropy in certain reversible deterministic nonequilibrium system, and its link with the collapse of such systems towards an attractor that is of lower dimension than the dimension of phase space. This means that in the steady state limit, the Gibbs entropy diverges to negative infinity. We argue that if the Gibbs entropy is expanded in a series involving 1, 2,... body terms, the divergence of the Gibbs entropy is manifest only in terms involving integrals whose dimension is higher than, approximately, the Kaplan–Yorke dimension of the steady state attractor. All the low order terms are finite and sum in the weak field limit to the local equilibrium entropy of linear irreversible thermodynamics.     

A novel four-wing chaotic attractor generated from a three-dimensional quadratic autonomous system

This paper presents a new 3D quadratic autonomous chaotic system which contains five system parameters and three quadratic cross-product terms, and the system can generate a single four-wing chaotic attractor with wide parameter ranges. Through theoretical analysis, the Hopf bifurcation processes are proved to arise at certain equilibrium points. Numerical bifurcation analysis shows that the system has many interesting complex dynamical behaviours; the system trajectory can evolve to a chaotic attractor from a periodic orbit or a fixed point as the proper parameter varies. Finally, an analog electronic circuit is designed to physically realize the chaotic system; the existence of four-wing chaotic attractor is verified by the analog circuit realization.     

Global control of spiral wave dynamics in an excitable domain of circular and elliptical shape

Experiments performed in a thin layer of the Belousov-Zhabotinsky solution subjected to a global feedback demonstrate the existence of the resonance attractor for meandering spiral waves within a domain of circular shape. In an elliptical domain, the resonance attractor can be destroyed due to a saddle-node bifurcation induced by a variation of the domain eccentricity. This conclusion explains the experimentally observed anchoring of spiral waves at certain points of an elliptical domain and is in good quantitative agreement with numerical data obtained for the Oregonator model.     

Long transients dynamics in biochemical networks

Summary  A Coupled Map Lattice, which simulates gene expression dynamics inside cells and cellular interactions on a regular lattice, shows a complex pattern of temporal behaviour. The model is represented as a network of genes interacting through their products in space and time in a lattice of genetically identical cells. Despite the fact that the system is described through a step function that imposes a simple repertoire of constant or oscillatory steady states, the dynamics over the lattice are extremely complex. One of the main feature of the asymptotic dynamics is the appearance of long transients in certain regions of parameter space, before the attainment of the final stable attractor. These dynamics, that can grow linearly or exponentially with lattice size, can become the only dynamics computationally observable. The study of the global dynamics-i.e. the average value of the variable over the lattice-shows a qualitative different behaviour depending on the region of the parameter space observed. In the case of the linear transient-growth region the system shows an average that falls quickly on a periodic attractor. In the exponential region values of the average quantities show a behaviour that has stochastic properties. At the boundary of these two regimes the system has an average that shows a complex behaviour before attainment of the final attractor. The possible implications of these results for the study of the dynamical aspects of gene regulation, biochemical pathways and in signal transduction in experimental systems are discussed. This work has been partially supported by CNR grant No. 95.01751.CT14 “Studio analitico della dinamica della regolazione genica e della morfogenesi#x201C;, and by funds from the National Ministry of Public Health. FB and RL would like to thank I.S.I., Torino, for the kind hospitality during the workshop of the EEC Network “Complexity and Chaos#x201D;, contract No. ERBCHRX-CT940546, in 1995 and 1996, during which part of this research has been done.