Introduction to Nonlinear Dynamics of Electronic Systems: Tutorial

The study of chaos has generated enormous interest in exploring the complexity of the behavior in nature and in technology. Many of the important features of chaotic dynamical systems can be seen using experimental and computational methods in simple nonlinear mechanical systems or electronic circuits. Starting with the study of a chaotic nonlinear mechanical system (driven damped pendulum) or a nonlinear electronic system (circuit Chua) we introduce the reader into the concepts of chaos order in Sharkovsky's sense, and topological invariants (topological entropy and topological frequencies). The Kirchhoff's circuit laws are a pair of laws that deal with the conservation of charge and energy in electric circuits, and the algebraic theory of graphs characterizes these linear systems in terms of cycles and cocycles (or cuts). Here we discuss methods (topological semiconjugacy to piecewise linear maps and Markov graphs) to find a similar situation for the nonlinear dynamics, to understanding chaotic dynamics. Thus to chaotic dynamics we associate a Markov graph, where the dynamical and topological invariants will be seen as graph theoretical quantities.     

Topological Fixed Point Theorems Do Not Hold for Random Dynamical Systems

The aim of this paper is to demonstrate that topological fixed point theorems have no canonical generalization to the case of random dynamical systems. This is done by using tools from algebraic ergodic theory. We give a criterion for the existence of invariant probability measures for group valued cocycles. With that, examples of continuous random dynamical systems on a compact interval without random invariant points, which are an appropriate generalization of fixed points, are constructed.     

Reducibility Results for Quasiperiodic Cocycles With Liouvillean Frequency

We study the reducibility problems for quasiperiodic cocycles in linear Lie groups with one frequency, irrespective of any Diophantine condition on the base dynamics. Under a non-degeneracy condition, a positive measure diagonalizable result is obtained for quasiperiodic GL(d,\mathbb R){GL(d,\mathbb R)} cocycles which are close to constants. It generalizes previous works by Avila–Fayad–Krikorian and Hou–You, and our approach is based on periodic approximation and KAM schemes.     

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezić, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.     

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS(Ⅱ)──PART 1

The study of linear and global properties of linear dynamical systems on vector bundles appeared rather extensive already in the past. Presently we propose to study perturbations of this linear dynamics. The perturbed dynamical system which we shall consider is no longer linear, while the properties to be studied will be still global in general. Moreover, we are intersted in the nonuniformly hyperbolic properties. In this paper, we set an appropriate definition for such perturbations. Though it appears somewhat not quite usual, yet has deeper root in standard systems of differential equations in the theory of differentiable dynamical systems. The general problem is to see which property of the original given by the dynamical system is persistent when a perturbation takes place. The whole content of the paper is devoted to establishing a theorem of this sort.     

Entropy of Dynamical Systems with Repetition Property

The repetition property of a dynamical system, a notion introduced in Boshernitzan and Damanik (Commun Math Phys 283:647–662, 2008), plays an importance role in analyzing spectral properties of ergodic Schrödinger operators. In this paper, entropy of dynamical systems with repetition property is investigated. It is shown that the topological entropy of dynamical systems with the global repetition property is zero. Minimal dynamical systems having both topological repetition property and positive topological entropy are constructed. This provides a class of ergodic Schrödinger operators with potentials generated by positive entropy minimal dynamical systems that, in contrast to common beliefs, admit no eigenvalues.     

Some asymptotic results for a class of stochastic systems with parametric excitations

Systems of stochastic ordinary differential equations dependent on a small parameter are studied. The equations are assumed to depend on two time scales: they are stochastic in a fast time t and they are deterministic in a slow time t. The method of analysis is based on a generalization of the Method of Averaging. Mathematical results are given valid for all t for sufficiently small. The mathematical results are applied to several examples of parametrically excited dynamical systems.     

To derive Lagrange equations of nonholonomic systems without using the Chetaev conditions

The present paper derives the equation of motion for a class of nonholonomic dynamical systems from the d’Alembert–Lagrange equation. In this paper, the relation dδ = δd is assumed only for those generalized coordinates whose variations are independent. For the remaining coordinates, the transpositional relations are derived by means of mathematics method, which are different from the Chetaev condition. Several examples are given.     

On {SL(2, \mathbb R)} Valued Smooth Proximal Cocycles and Cocycles with Positive Lyapunov Exponents Over Irrational Rotation Flows

Consider the class of C ^r -smooth SL(2, \mathbb R){SL(2, \mathbb R)} valued cocycles, based on the rotation flow on the two torus with irrational rotation number α. We show that in this class, (i) cocycles with positive Lyapunov exponents are dense and (ii) cocycles that are either uniformly hyperbolic or proximal are generic, if α satisfies the following Liouville type condition: |a-\fracp_nq_n| ￡ C exp (-q^r+1+k_n)\left|\alpha-\frac{p_n}{q_n}\right| \leq C {\rm exp} (-q^{r+1+\kappa}_{n}), where C >  0 and 0 < k < 1{0 < \kappa <1 } are some constants and \fracP_nq_n{\frac{P_n}{q_n}} is some sequence of irreducible fractions.     

Rich dynamics caused by diffusion

It is an awfully difficult task to design an efficient numerical method for bifurcation diagrams, the graphs of Lyapunov exponents, or the topological entropy about discrete dynamical systems by linear/nonlinear diffusion with the Direchlet/Neumann- boundary conditions. Until now there are less works concerned with such a problem. In this paper, we propose a scheme about bifurcating analysis in a series of discrete-time dynamical systems with linear/nonlinear diffusion terms under the periodic boundary conditions. The complexity of dynamical behaviors caused by the diffusion term are to be determined. Bifurcation diagrams are shown by numerical simulation and chaotic behavior (chaotic Turing patterns) is demonstrated by computing the largest Lyapunov exponent. Our theoretical model can give an interesting case study about the phenomenon: the individuals exhibit a very simple dynamics but the groups with linear/nonlinear coupling can own a complex dynamics including fluctuation, periodicity and even chaotic behavior. We find that diffusion can trigger chaotic behavior in the present system and there exist multiple Turing patterns. It is interesting as regular or chaotic patterns can be reported in this study. Chaotic orbits emerge when exploring further in the diffusion coefficient space, and such a behavior is entirely absent in the corresponding continuous time-space system. The method proposed in the present paper is innovative and the conclusion is novel.

     

On the Hartman–Grobman Theorem with Parameters

The Hartman–Grobman Theorem of linearization is extended to families of dynamical systems in a Banach space \mathbb X{\mathbb X} , depending continuously on parameters. We prove that the conjugacy also changes continuously. The cases of nonlinear maps and flows are considered, and both in global and local versions, but global in the parameters. To use a special version of the Banach–Caccioppoli Theorem we introduce equivalent norms on \mathbb X{\mathbb X} depending on the parameters. The functional setting is suitable for applications to some nonlinear evolution partial differential equations like the nonlinear beam equation.     

General hyperbolic difference formulas for linear and quasilinear hyperbolic equations

The method of non-standard finite elements was used to develop multilevel difference schemes for linear and quasilinear hyperbolic equations with Dirichlet boundary conditions. A closed form equation of kth-order accuracy in space and time (O(Δt^k, Δx^k)) was developed for one-dimensional systems of linear hyperbolic equations with Dirichlet boundary conditions. This same equation is also applied to quasilinear systems. For the quasilinear systems a simple iteration technique was used to maintain the kth-order accuracy. Numerical results are presented for the linear and non-linear inviscid Burger's equation and a system of shallow water equations with Dirichlet boundary conditions.     

Nonlinear Stability for Multidimensional Fourth-Order Shock Fronts

We consider the question of stability for planar wave solutions that arise in multidimensional conservation laws with only fourth-order regularization. Such equations arise, for example, in the study of thin films, for which planar waves correspond to fluid coating a pre-wetted surface. An interesting feature of these equations is that both compressive, and undercompressive, planar waves arise as solutions (compressive or undercompressive with respect to asymptotic behavior relative to the un-regularized hyperbolic system), and numerical investigation by Bertozzi, Münch, and Shearer indicates that undercompressive waves can be nonlinearly stable. Proceeding with pointwise estimates on the Green's function for the linear fourth-order convection–regularization equation that arises upon linearization of the conservation law about the planar wave solution, we establish that under general spectral conditions, such as appear to hold for shock fronts arising in our motivating thin films equations, compressive waves are stable for all dimensions d≧2 and undercompressive waves are stable for dimensions d≧3. (In the special case d=1, compressive waves are stable under a very general spectral condition.) We also consider an alternative spectral criterion (valid, for example, in the case of constant-coefficient regularization), for which we can establish nonlinear stability for compressive waves in dimensions d≧3 and undercompressive waves in dimensions d≧5. The case of stability for undercompressive waves in the thin films equations for the critical dimensions d=1 and d=2 remains an interesting open problem.     

A time-domain nonlinear system identification method based on multiscale dynamic partitions

Based on a theoretical foundation for empirical mode decomposition, which dictates the correspondence between the analytical and empirical slow-flow analyses, we develop a time-domain nonlinear system identification (NSI) technique. This NSI method is based on multiscale dynamic partitions and direct analysis of measured time series, and makes no presumptions regarding the type and strength of the system nonlinearity. Hence, the method is expected to be applicable to broad classes of applications involving time-variant/time-invariant, linear/nonlinear, and smooth/non-smooth dynamical systems. The method leads to nonparametric reduced order models of simple form; i.e., in the form of coupled or uncoupled oscillators with time-varying or time-invariant coefficients forced by nonhomogeneous terms representing nonlinear modal interactions. Key to our method is a slow/fast partition of transient dynamics which leads to the identification of the basic fast frequencies of the dynamics, and the subsequent development of slow-flow models governing the essential dynamics of the system. We provide examples of application of the NSI method by analyzing strongly nonlinear modal interactions in two dynamical systems with essentially nonlinear attachments.     

NOTES ON A STUDY OF VECTOR BUNDLE DYNAMICAL SYSTEMS(Ⅱ)──PART 1

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Motion of an Elastic Solid inside an Incompressible Viscous Fluid

The motion of an elastic solid inside an incompressible viscous fluid is ubiquitous in nature. Mathematically, such motion is described by a PDE system that couples the parabolic and hyperbolic phases, the latter inducing a loss of regularity which has left the basic question of existence open until now.In this paper, we prove the existence and uniqueness of such motions (locally in time), when the elastic solid is the linear Kirchhoff elastic material. The solution is found using a topological fixed-point theorem that requires the analysis of a linear problem consisting of the coupling between the time-dependent Navier-Stokes equations set in Lagrangian variables and the linear equations of elastodynamics, for which we prove the existence of a unique weak solution. We then establish the regularity of the weak solution; this regularity is obtained in function spaces that scale in a hyperbolic fashion in both the fluid and solid phases. Our functional framework is optimal, and provides the a priori estimates necessary for us to employ our fixed-point procedure.This revised version was published in April 2005. The volume number has now been inserted into the citation line.     

Countable Compacta Admitting Homeomorphisms with Positive Sequence Entropy

Let (X, T) be a topological dynamical system (TDS), Ω(T) be the set of non-wondering points and be the topological sequence entropy. In this paper, an example on a countable compactum X with is given. Then for TDSs on countable compacta X, it is proved that when d(X) ≤ 1, ; and when d(X) ≥ 2, there exists a homeomorphism T on X such that X ^d is the sequence entropy set of (X, T), where d(X) and X ^d are the derived degree of X and the set of all accumulation points of X respectively. Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

PRICE: primitive centred schemes for hyperbolic systems

We present first‐ and higher‐order non‐oscillatory primitive (PRI) centred (CE) numerical schemes for solving systems of hyperbolic partial differential equations written in primitive (or non‐conservative) form. Non‐conservative systems arise in a variety of fields of application and they are adopted in that form for numerical convenience, or more importantly, because they do not posses a known conservative form; in the latter case there is no option but to apply non‐conservative methods. In addition we have chosen a centred, as distinct from upwind, philosophy. This is because the systems we are ultimately interested in (e.g. mud flows, multiphase flows) are exceedingly complicated and the eigenstructure is difficult, or very costly or simply impossible to obtain. We derive six new basic schemes and then we study two ways of extending the most successful of these to produce second‐order non‐oscillatory methods. We have used the MUSCL‐Hancock and the ADER approaches. In the ADER approach we have used two ways of dealing with linear reconstructions so as to avoid spurious oscillations: the ADER TVD scheme and ADER with ENO reconstruction. Extensive numerical experiments suggest that all the schemes are very satisfactory, with the ADER/ENO scheme being perhaps the most promising, first for dealing with source terms and secondly, because higher‐order extensions (greater than two) are possible. Work currently in progress includes the application of some of these ideas to solve the mud flow equations. The schemes presented are generic and can be applied to any hyperbolic system in non‐conservative form and for which solutions include smooth parts, contact discontinuities and weak shocks. The advantage of the schemes presented over upwind‐based methods is simplicity and efficiency, and will be fully realized for hyperbolic systems in which the provision of upwind information is very costly or is not available. Copyright © 2003 John Wiley & Sons, Ltd.     

The global analysis of higher order nonlinear dynamical systems and the application of cell-to-cell mapping method

In this paper, the general characteristics and the topological consideration of the global behaviors of higher order nonlinear dynamical systems and the characteristics of the application of cell-to-cell mapping method in this analysis are expounded. Specifically, the global analysis of a system of two weakly coupled van der Pol oscillators using cell-to-cell mapping method is presented.The analysis shows that for this system, there exist two stable limit cycles in 4-dimensional state space, and the whole 4-dimensional state space is divided into two almost equal parts which are, respectively, the two asymototically stable domains of attraction of the two periodic motions of the two stable limit cycles. The validities of these conclusions about the global behaviors are also verified by direct long term numerical integration. Thus, it can be seen that the cell-to-cell mapping method for global analysis of fourth order nonlinear dynamical systems is quite effective.     

Transversality for Cyclic Negative Feedback Systems

Transversality of stable and unstable manifolds of hyperbolic periodic trajectories is proved for monotone cyclic systems with negative feedback. Such systems in general are not in the category of monotone dynamical systems in the sense of Hirsch. Our main tool utilized in the proofs is the so-called cone of high rank. We further show that stable and unstable manifolds between a hyperbolic equilibrium and a hyperbolic periodic trajectory, or between two hyperbolic equilibria with different dimensional unstable manifolds also intersect transversely.