Generalizations of SRB Measures to Nonautonomous,Random, and Infinite Dimensional Systems

We review some developments that are direct outgrowths of, or closely related to, the idea of SRB measures as introduced by Sinai, Ruelle and Bowen in the 1970s. These new directions of research include the emergence of strange attractors in periodically forced dynamical systems, random attractors in systems defined by stochastic differential equations, SRB measures for infinite dimensional systems including those defined by large classes of dissipative PDEs, quasi-static distributions for slowly varying time-dependent systems, and surviving distributions in leaky dynamical systems.     

Computation in gene networks

Genetic regulatory networks have the complex task of controlling all aspects of life. Using a model of gene expression by piecewise linear differential equations we show that this process can be considered as a process of computation. This is demonstrated by showing that this model can simulate memory bounded Turing machines. The simulation is robust with respect to perturbations of the system, an important property for both analog computers and biological systems. Robustness is achieved using a condition that ensures that the model equations, that are generally chaotic, follow a predictable dynamics.     

Time-Dependent First Integrals,Nonlinear Dynamical Systems,and Numerical Integration

Many nonlinear dynamical systems expressed as autonomous systems of first-order ordinary differential equations admit first integrals and explicitly time-dependent first integrals. Under numerical integration these first integrals should be preserved. We discuss this case for explicitly time-dependent first integrals.     

Fractional order chaotic systems: history,achievements, applications,and future challenges

Motivated by the importance of study on the complex behaviors, which may be exhibited by fractional order differential equations, this review paper focuses on dynamical fractional order systems exhibiting chaotic behaviors. The review begins with a brief history on the first publications on the above-mentioned subject. Then, the review is continued by investigating the recent progresses relevant to fractional order chaotic systems. Furthermore, a summary on some applications for such systems, which have been reported in the literature, is presented. Finally, the paper is closed by discussing some open problems on the aforementioned research subject. These open problems, as future challenges for further study on fractional order chaotic systems, can specify some direction lines for continuing the research on that subject.

     

A new chaotic system without linear term and its impulsive synchronization

In this paper, a new chaotic system without linear term is presented by ordinary differential equations, and some basic dynamical properties are studied. Moreover, the impulsive synchronization problem of the new chaotic systems is investigated by using uncertain impulsive control matrix. Using the impulsive theory, sufficient conditions are derived for the synchronization of new chaotic systems. Numerical simulation example is provided to verify the effectiveness of the proposed approach.     

场方法的改进及其在积分Riemann-Cartan空间运动方程中的应用

和Hamilton-Jacobi方法类似,Vujanovi?场方法把求解常微分方程组特解的问题转化为寻找一个一阶拟线性偏微分方程(基本偏微分方程)完全解的问题,但Vujanovi?场方法依赖于求出基本偏微分方程的完全解,而这通常是困难的,这就极大地限制了场方法的应用.本文将求解常微分方程组特解的Vujanovi?场方法改进为寻找动力学系统运动方程第一积分的场方法,并将这种方法应用于一阶线性非完整约束系统Riemann-Cartan位形空间运动方程的积分问题中.改进后的场方法指出,只要找到基本偏微分方程的包含m(m≤ n,n为基本偏微分方程中自变量的数目)个任意常数的解,就可以由此找到系统m个第一积分.特殊情况下,如果能够求出基本偏微分方程的完全解(完全解是m=n时的特例),那么就可以由此找到≤系统全部第一积分,从而完全确定系统的运动.Vujanovi?场方法等价于这种特殊情况.     

Control of noisy chaotic motion in a system with nonlinear excitation and restoring forces

In this study we examine the complex and chaotic oscillations of a dynamical system with nonlinear excitation and restoring forces for the purpose of controlling these oscillatory states. The physical system, modeled as a system of first-order nonlinear ordinary differential equations, takes into account a geometric nonlinearity in the restoring force, a quadratic viscous drag, and a harmonic excitation force. It is controlled using small perturbations about a selected unstable cycle and control is instigated for periodic cycles of varying periodicities. The controller, when applied on the dynamical system with additive random noise in the excitation, successfully controls the system with noise levels in excess of 5% of the total energy, giving the first evidence that (stochastic) control of these systems is possible. (c) 1997 American Institute of Physics.     

Approximate controllability of nonlinear impulsive differential systems

Many practical systems in physical and biological sciences have impulsive dynamical be- haviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability issue for nonlinear impulsive differential and neutral functional differential equations in Hilbert spaces. Based on the semigroup theory and fixed point approach, sufficient conditions for approximate controllability of impulsive differential and neutral functional differential equations are established. Finally, two examples are presented to illustrate the utility of the proposed result. The results improve some recent results.     

Topological Field Theory and Computing with Instantons

It is well known that dynamical systems may be employed as computing machines. However, not all dynamical systems offer particular advantages compared to the standard paradigm of computation, in regard to efficiency and scalability. Recently, it was suggested that a new type of machines, named digital –hence scalable– memcomputing machines (DMMs), that employ non‐linear dynamical systems with memory, can solve complex Boolean problems efficiently. This result was derived using functional analysis without, however, providing a clear understanding of which physical features make DMMs such an efficient computational tool. Here, we show, using recently proposed topological field theory of dynamical systems, that the solution search by DMMs is a composite instanton. This process effectively breaks the topological supersymmetry common to all dynamical systems, including DMMs. The emergent long‐range order – a collective dynamical behavior– allows logic gates of the machines to correlate arbitrarily far away from each other, despite their non‐quantum character. We exemplify these results with the solution of prime factorization, but the conclusions generalize to DMMs applied to any other Boolean problem.     

A STOCHASTIC OPTIMAL SEMI-ACTIVE CONTROL STRATEGY FOR ER/MR DAMPERS

A stochastic optimal semi-active control strategy for randomly excited systems using electrorheological/magnetorheological (ER/MR) dampers is proposed. A system excited by random loading and controlled by using ER/MR dampers is modelled as a controlled, stochastically excited and dissipated Hamiltonian system with n degrees of freedom. The control forces produced by ER/MR dampers are split into a passive part and an active part. The passive control force is further split into a conservative part and a dissipative part, which are combined with the conservative force and dissipative force of the uncontrolled system, respectively, to form a new Hamiltonian and an overall passive dissipative force. The stochastic averaging method for quasi-Hamiltonian systems is applied to the modified system to obtain partially completed averaged Itô stochastic differential equations. Then, the stochastic dynamical programming principle is applied to the partially averaged Itô equations to establish a dynamical programming equation. The optimal control law is obtained from minimizing the dynamical programming equation subject to the constraints of ER/MR damping forces, and the fully completed averaged Itô equations are obtained from the partially completed averaged Itô equations by replacing the control forces with the optimal control forces and by averaging the terms involving the control forces. Finally, the response of semi-actively controlled system is obtained from solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov equation associated with the fully completed averaged Itô equations of the system. Two examples are given to illustrate the application and effectiveness of the proposed stochastic optimal semi-active control strategy.     

The Few-Body Approximation in Nuclear,Atomic, and Molecular Physics

The quantum theory of few-body scattering based on Faddeev-Yakubovsky integral and differential equa¬tions is applied to calculations of various processes (elastic, inelastic, atom exchange, and dissociative) in nuclear, atomic, and molecular physics. Analytical solutions of these equations are presented for various limiting cases. The methods used for solving the integral and differential systems of equations are discussed.     

A data-analysis method for identifying differential effects of time-delayed feedback forces and periodic driving forces in stochastic systems

In this work a method is developed for analyzing time series of periodically driven stochastic systems involving time-delayed feedback. The proposed data-analysis method yields dynamical models in terms of stochastic delay differential equations. On the basis of these dynamical models differential effects of driving forces and time-delayed feedback forces can be identified.     

Controlling a time-delay system using multiple delay feedback control

In this paper multiple delay feedback control (MDFC) with different and independent delay times is shown to be an efficient method for stabilizing fixed points in finite-dimensional dynamical systems. Whether MDFC can be applied to infinite-dimensional systems has been an open question. In this paper we find that for infinite-dimensional systems modelled by delay differential equations, MDFC works well for stabilizing (unstable) steady states in long-, moderate- and short-time delay regions, in particular for the hyperchaotic case.     

HOW TO EXTEND ANY DYNAMICAL SYSTEM SO THAT IT BECOMES ISOCHRONOUS,ASYMPTOTICALLY ISOCHRONOUS OR MULTI-PERIODIC

We indicate how one can extend any dynamical system (namely, any system of nonlinearly coupled autonomous ordinary differential equations) so that the extended dynamical system thereby obtained is either isochronous or asymptotically isochronous or multi-periodic, namely its generic solutions are either completely periodic with a fixed period or tend asymptotically, in the remote future, to such completely periodic functions or are multi-periodic (or become multi-periodic only asymptotically, in the remote future). In all cases the scale of the periodicity can be arbitrarily assigned. Moreover, the solutions of the extended systems are generally well approximated by those of the original, unmodified, systems, up to a constant rescaling of the independent variable (time), as long as their evolution is considered over time intervals short with respect to the (arbitrarily assigned) periodicities characterizing the extended systems. Several examples are displayed. In some cases the general solution of these dynamical systems is also exhibited; in others, this is impossible inasmuch as the models being manufactured are extensions of dynamical systems displaying chaotic evolutions, such as, for instance, the well-known Lorenz model of 3 nonlinearly coupled ODEs.     

Dielectric constant of graphene-on-polarized substrate: A tight-binding model study

By using the bifurcation theory of planar dynamical systems and the qualitative theory of differential equations, we studied the dynamical behaviours and exact travelling wave solutions of the modified generalized Vakhnenko equation (mGVE). As a result, we obtained all possible bifurcation parametric sets and many explicit formulas of smooth and non-smooth travelling waves such as cusped solitons, loop solitons, periodic cusp waves, pseudopeakon solitons, smooth periodic waves and smooth solitons. Moreover, we provided some numerical simulations of these solutions.     

Identifying parameter by identical synchronization between different systems

In this paper, parameters of a given (chaotic) dynamical system are estimated from time series by using identical synchronization between two different systems. This technique is based on the invariance principle of differential equations, i.e., a dynamical Lyapunov function involving synchronization error and the estimation error of parameters. The control used in this synchronization consists of feedback and adaptive control loop associated with the update law of estimation parameters. Our estimation process indicates that one may identify dynamically all unknown parameters of a given (chaotic) system as long as time series of the system are available. Lorenz and Rossler systems are used to illustrate the validity of this technique. The corresponding numerical results and analysis on the effect of noise are also given.     

A unifying definition of synchronization for dynamical systems

We propose a unifying definition for synchronization between stationary finite dimensional deterministic dynamical systems. By example, we show that the synchronization phenomena discussed in the dynamical systems literature fits within the framework of this definition, and discuss problems with previous definitions of synchronization. We conclude with a discussion of possible extensions of the definition to infinite dimensional systems described by partial differential equations and/or systems where noise is present. (c) 2000 American Institute of Physics.     

Nonlinear,nondispersive wave equations: Lagrangian and Hamiltonian functions in the hodograph transformation

The hodograph transformation is generally used in order to associate a system of linear partial differential equations to a system of nonlinear (quasilinear) differential equations by interchanging dependent and independent variables. Here we consider the case when the nonlinear differential system can be derived from a Lagrangian density and revisit the hodograph transformation within the formalism of the Lagrangian-Hamiltonian continuous dynamical systems.Restricting to the case of nondissipative, nondispersive one-dimensional waves, we show that the hodograph transformation leads to a linear partial differential equation for an unknown function that plays the role of the Lagrangian in the hodograph variables. We then define the corresponding hodograph Hamiltonian and show that it turns out to coincide with the wave amplitude. i.e., with the unknown function of the independent variables to be solved for in the initial nonlinear wave equation.     

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves*

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new $(2+1)$-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.     

Identifying and comparing states of time-delayed systems: phase diagrams and applications to human motor control systems

A data driven characterization of time-delayed stochastic systems is proposed in terms of linear delay differential equations and two drift parameters. It is shown how these parameters determine the states of such systems with respect to generalized phase diagrams. This approach allows for a comparison of systems with different parameters as exemplified for two motor control tasks: tracking and force production.