Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

Single Parameter Dissipativity and Attractors in DiscreteTirne Asynchronous Systems ∗

Discrete time nonautonomous dynamical systems generated by nonautonomous difference equations are formulated as discrete time skew—product systems consisting of cocycle state mappings that are driven by discrete time autonomous dynamical systems. Forwards and pullback attractors are two possible generalizations of autonomous attractors to such systems. Their existence follows from appropriate forwards or pullback dissipativity conditions. For discrete time nonautonomous dynamical systems generated by asynchronous systems with frequency updating components such a dissipativity condition is usually known for a single starting parameter value of the driving system. Additional conditions that then ensure the existence of a forwards or pullback attractor for such an asynchronous system are investigated here     

A numerical method with particle conservation for the Maxwell-Dirac system

A numerical method is presented for solving the Maxwell-Dirac systems. The Maxwell equations with particle and current densities as the source terms are discretized explicitly. To guarantee the particle conservation, the Dirac equations coupled electromagnetic potentials are discretized by the time-splitting method and implicit finite difference. These numerical schemes are conservative in particle density and have second-order accuracy in time and space. One-dimensional numerical results are given to validate the accuracy and the conservation and three-dimensional examples are presented to describe dynamical behaviors of the Maxwell-Dirac system with several external potentials.     

Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations, including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations.     

Sigma map dynamics and bifurcations

Some interesting variants of walking droplet based discrete dynamical bifurcations arising from diffeomorphisms are analyzed in detail. A notable feature of these new bifurcations is that, like Smale horseshoes, they can be represented by simple geometric paradigms, which markedly simplify their analysis. The two-dimensional diffeomorphisms that produce these bifurcations are called sigma maps or double sigma maps for reasons that are made manifest in this investigation. Several examples are presented along with their dynamical simulations.     

Formal diagonalization of a discrete lax operator and conservation laws and symmetries of dynamical systems

We consider the problem of constructing a formal asymptotic expansion in the spectral parameter for an eigenfunction of a discrete linear operator. We propose a method for constructing an expansion that allows obtaining conservation laws of discrete dynamical systems associated with a given linear operator. As illustrative examples, we consider known nonlinear models such as the discrete potential Kortewegde Vries equation, the discrete version of the derivative nonlinear Schrödinger equation, the Veselov-Shabat dressing chain, and others. We describe the infinite set of conservation laws for the discrete Toda chain corresponding to the Lie algebra A ₁ ⁽¹⁾ . We find new examples of integrable systems of equations on a square lattice.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

Standard Systems of Discrete Differentiable Dynamical Systems

The theory of Liao standard systems of difference equations for discrete differentiable dynamical systems over compact manifolds has been developed. Some relations between the topological structures of their phase portraits and that of their corresponding linear systems have been presented as well.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

Discrete and continuous coupled nonlinear integrable systems via the dressing method

A discrete analog of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrödinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one‐directional linear wave equations, an appropriate matrix Riemann‐Hilbert problem is constructed, and a discrete matrix nonlinear Schrödinger system of equations is derived, together with its Lax pair. The corresponding compatible vector reductions admitted by these systems are also discussed, as well as their continuum limits. Finally, by increasing the size of the problem, three‐component discrete and continuous integrable discrete systems are derived, as well as their generalizations to systems with an arbitrary number of components.     

Ellipsoidal approximation of attainability sets of a linear system with indeterminate matrix

Linear dynamical systems described by finite-difference or differential equations are considered. It is assumed that the matrix of the system is either completely known or is subject to uncontrollable perturbations, so that each element is known only to within a certain possible interval. Outer approximations, by means of ellipsoids, are constructed for the attainability sets of such systems. The equations of evolution of the approximating ellipsoids are obtained. An example is presented.     

Continuous and Discrete Clebsch Variational Principles

The Clebsch method provides a unifying approach for deriving variational principles for continuous and discrete dynamical systems where elements of a vector space are used to control dynamics on the cotangent bundle of a Lie group via a velocity map. This paper proves a reduction theorem which states that the canonical variables on the Lie group can be eliminated, if and only if the velocity map is a Lie algebra action, thereby producing the Euler–Poincaré (EP) equation for the vector space variables. In this case, the map from the canonical variables on the Lie group to the vector space is the standard momentum map defined using the diamond operator. We apply the Clebsch method in examples of the rotating rigid body and the incompressible Euler equations. Along the way, we explain how singular solutions of the EP equation for the diffeomorphism group (EPDiff) arise as momentum maps in the Clebsch approach. In the case of finite-dimensional Lie groups, the Clebsch variational principle is discretized to produce a variational integrator for the dynamical system. We obtain a discrete map from which the variables on the cotangent bundle of a Lie group may be eliminated to produce a discrete EP equation for elements of the vector space. We give an integrator for the rotating rigid body as an example. We also briefly discuss how to discretize infinite-dimensional Clebsch systems, so as to produce conservative numerical methods for fluid dynamics.      

On the global periodicity of discrete dynamical systems and application to rational difference equations

A new necessary condition for global periodicity of discrete dynamical systems and of difference equations is obtained here. This condition will be applied to contribute to solving the problem of global periodicity for second order rational difference equations.     

A Darboux-type Theory of Integrability for Discrete Dynamical Systems

There is a method for searching first integrals for polynomial ordinary differential equations (usually called Darboux method) based on the knowledge of several of their invariant algebraic hypersurfaces. We extend this method to discrete dynamical systems, providing a way of searching invariants for them and we give several examples of application.     

Conserved Quantities and Symmetries Related to Stochastic Dynamical Systems

The present article focuses on the three topics related to the notions of "conserved quantities" and "symmetries" in stochastic dynamical systems described by stochastic differential equations of Stratonovich type. The first topic is concerned with the relation between conserved quantities and symmetries in stochastic Hamilton dynamical systems, which is established in a way analogous to that in the deterministic Hamilton dynamical theory. In contrast with this, the second topic is devoted to investigate the procedures to derive conserved quantities from symmetries of stochastic dynamical systems without using either the Lagrangian or Hamiltonian structure. The results in these topics indicate that the notion of symmetries is useful for finding conserved quantities in various stochastic dynamical systems. As a further important application of symmetries, the third topic treats the similarity method to stochastic dynamical systems. That is, it is shown that the order of a stochastic system can be reduced, if the system admits symmetries. In each topic, some illustrative examples for stochastic dynamical systems and their conserved quantities and symmetries are given.     

Evaluation of the largest Lyapunov exponent in dynamical systems with time delay

The method of estimation of the largest Lyapunov exponents for dynamical systems with time delay has been developed. This method can be applied both for flows and discrete maps. Our approach is based on the phenomenon of synchronization of identical systems coupled by linear negative feedback mechanism (flows) and exponential perturbation (maps). The existence of linear dependence of the largest Lyapunov exponent on the coupled parameter allows the precise estimation of this exponent.     

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.     

Discrete mechanics and its application to the solution of the n-body problem

A theory of discrete mechanics is developed based on the results of D. Greenspan. Discrete dynamical equations in an inertial frame, in a coordinate system related to some material point, and in a rotating frame are given and the consistency, stability, and convergence of the methods are studied and some numerical examples presented.     

Identification of Parameters of Nonlinear Dynamical Systems Simulated by Volterra Polynomials

We study the identification methods for the nonlinear dynamical systems described by Volterra series. One of the main problems in the dynamical system simulation is the problem of the choice of the parameters allowing the realization of a desired behavior of the system. If the structure of the model is identified in advance, then the solution to this problem closely resembles the identification problem of the system parameters. We also investigate the parameter identification of continuous and discrete nonlinear dynamical systems. The identification methods in the continuous case are based on application of the generalized Borel Theorem in combination with integral transformations. To investigate discrete systems, we use a discrete analog of the generalized Borel Theorem in conjunction with discrete transformations. Using model examples, we illustrate the application of the developed methods for simulation of systems with specified characteristics.     

High-order adaptive Gegenbauer integral spectral element method for solving non-linear optimal control problems

In this work, we propose an adaptive spectral element algorithm for solving non-linear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer–Gauss points combined with very accurate and stable numerical quadratures to fully discretize the multiple-phase integral form of the optimal control problem. The proposed algorithm relies on exploiting the underlying smoothness properties of the solutions for computing approximate solutions efficiently. In particular, the method brackets discontinuities and ‘points of nonsmoothness’ through a novel local adaptive algorithm, which achieves a desired accuracy on the discrete dynamical system equations by adjusting both the mesh size and the degree of the approximating polynomials. A rigorous error analysis of the developed numerical quadratures is presented. Finally, the efficiency of the proposed method is demonstrated on three test examples from the open literature.