Lyapunov exponents and invariant measures on a projective bundle

Mathematical Notes - A discrete dynamical system generated by a diffeomorphism f on a compact manifold is considered. The Morse spectrum is the limit set of Lyapunov exponents of periodic...     

Topological conjugacy of constant length substitution dynamical systems

Primitive constant length substitutions generate minimal symbolic dynamical systems. In this article we present an algorithm which can produce the list of injective substitutions of the same length that generate topologically conjugate systems. We show that each conjugacy class contains infinitely many substitutions which are not injective. As examples, the Toeplitz conjugacy class contains three injective substitutions (two on two symbols and one on three symbols), and the length two Thue–Morse conjugacy class contains twelve substitutions, among which are two on six symbols. Together, they constitute a list of all primitive substitutions of length two with infinite minimal systems which are factors of the Thue–Morse system.     

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.     

Tilings,substitution systems and dynamical systems generated by them

The object of this work is to study the properties of dynamical systems defined by tilings. A connection to symbolic dynamical systems defined by one- and two-dimensional substitution systems is shown. This is used in particular to show the existence of a tiling system such that its corresponding dynamical system is minimal and topological weakly mixing. We remark that for one-dimensional tilings the dynamical system always contains periodic points.     

Topological conjugacy to given constant length substitution minimal systems

We find necessary and sufficient conditions for a symbolic dynamical system to be topologically conjugate to any given constant length substitution minimal system, thus extending the results in Coven et al. (2008) for the Morse and Toeplitz substitutions.     

Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries

In this paper, we define a relative Morse index for two continuous symmetric matrices paths in R2n satisfying condition (B1) and study its relation with the Maslov-type indices under brake orbit boundary value of these two symmetric matrices paths. As applications, using this relation we obtain a multiple existence of periodic brake orbit solutions of asymptotically linear Hamiltonian system in the presence of symmetries.     

Generalized Morse sequences

Summary A method for construction of almost periodic points in the shift space on two symbols is developed, and a necessary and sufficient condition is given for the orbit closure of such a point to be strictly ergodic. Points satisfying this condition are called generalized Morse sequences. The spectral properties of the shift operator in strictly ergodic systems arising from generalized Morse sequences are investigated. It is shown that under certain broad regularity conditions both the continuous and discrete parts of the spectrum are non-trivial. The eigenfunctions and eigenvalues are calculated. Using the results, given any subgroup of the group of roots of unity, a generalized Morse sequence can be constructed whose continuous spectrum is non-trivial and whose eigenvalue group is precisely the given group. New examples are given for almost periodic points whose orbit closure is not strictly ergodic.     

Conley index condition for asymptotic stability

In this paper, we use Conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time systems. The Conley index is a topological generalization of the Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems.     

Numerical Continuation of Hamiltonian Relative Periodic Orbits

The bifurcation theory and numerics of periodic orbits of general dynamical systems is well developed, and in recent years, there has been rapid progress in the development of a bifurcation theory for dynamical systems with structure, such as symmetry or symplecticity. But as yet, there are few results on the numerical computation of those bifurcations. The methods we present in this paper are a first step toward a systematic numerical analysis of generic bifurcations of Hamiltonian symmetric periodic orbits and relative periodic orbits (RPOs). First, we show how to numerically exploit spatio-temporal symmetries of Hamiltonian periodic orbits. Then we describe a general method for the numerical computation of RPOs persisting from periodic orbits in a symmetry breaking bifurcation. Finally, we present an algorithm for the numerical continuation of non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair. Our path following algorithm is based on a multiple shooting algorithm for the numerical computation of periodic orbits via an adaptive Poincaré section and a tangential continuation method with implicit reparametrization. We apply our methods to continue the famous figure eight choreography of the three-body system. We find a relative period doubling bifurcation of the planar rotating eight family and compute the rotating choreographies bifurcating from it.      

Tracking control of chaotic spinning disks via nonlinear dynamic output feedback with input constraints

In many control engineering applications, it is impossible or expensive to measure all the states of the dynamical system and only the system output is available for controller design. In this study, a new dynamic output feedback control algorithm is proposed to stabilize the unstable periodic orbit of chaotic spinning disks with incomplete state information. The proposed control structure is based on the T‐S fuzzy systems. This investigation also introduces a new design procedure to satisfy a constraint on the T‐S fuzzy dynamic output feedback control signal. This procedure is independent of the exact value of initial states. Finally, computer simulations are accomplished to illustrate the performance of the proposed control algorithm. © 2015 Wiley Periodicals, Inc. Complexity 21: 148–159, 2016     

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

     

Morse equation of attractors for nonsmooth dynamical systems

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.     

On invariant closed curves for one-step methods

Summary We show that a one-step method as applied to a dynamical system with a hyperbolic periodic orbit, exhibits an invariant closed curve for sufficiently small step size. This invariant curve converges to the periodic orbit with the order of the method and it inherits the stability of the periodic orbit. The dynamics of the one-step method on the invariant curve can be described by the rotation number for which we derive an asymptotic expression. Our results complement those of [2, 3] where one-step methods were shown to create invariant curves if the dynamical system has a periodic orbit which is stable in either time direction or if the system undergoes a Hopf bifurcation.     

Dichotomy spectra and Morse decompositions of linear nonautonomous differential equations

Recently, the existence of Morse decompositions for nonautonomous dynamical systems was shown for three different time domains: the past, the future and—in the linear case—the entire time. In this article, notions of exponential dichotomy are discussed with respect to the three time domains. It is shown that an exponential dichotomy gives rise to an attractor-repeller pair in the projective space, which is a building block of a Morse decomposition. Moreover, based on the notions of exponential dichotomy, dichotomy spectra are introduced, and it is proved that the corresponding spectral manifolds lead to Morse decompositions in the projective space.     

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.     

A spectral theory for linear differential systems

This paper is concerned with continuous and discrete linear skew-product dynamical systems including those generated by linear time-varying ordinary differential equations. The concept of spectrum is introduced for a linear skew-product dynamical system. In the case of a system of ordinary differential equations with constant coefficients the spectrum reduces to the real parts of the eigenvalues. In the general case continuous spectrum can occur and under certain conditions it consists of finitely many compact intervals of the real line, their number not exceeding the dimension of the system. A spectral decomposition theorem is proved which says that a certain naturally defined vector bundle is the sum of invariant subbundles, each one associated with a spectral subinterval. This partially generalizes the Jordan decomposition in the case of constant coefficients. A perturbation theorem is proved which says that nearby systems have spectra which are close. Almost periodic systems are given special attention.     

The Morse index,repeller-attractor pairs and the connection index for semiflows on noncompact spaces

This paper extends the Morse index theory of C. C. Conley to semiflows π on a noncompact meric space X. π is assumed to satisfy a hypothesis related to conditional α-contraction. We collect background material, define quasi-index pairs and the Morse index of a compact, isolated invariant set K, and prove that the Morse index is a connected simple system. We study repeller-attractor pairs in K, define index triples, and prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, we consider paths (continuous families) of pairs (π, K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the author's previous work: On the homotopy index for infinite-dimensional semiflows (Trans. Amer. Math. Soc.269 (1982), 351–382).     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.     

“Invariance-Inducing” Control of Nonlinear Discrete-Time Dynamical Systems

The present study aims at the derivation of model-based control laws that attain the invariance objective for nonlinear skew-product discrete-time dynamical systems. The problem under consideration naturally arises in a variety of control problems pertaining to physical/chemical systems, and in the present study, it is conveniently formulated and addressed in the context of functional equations theory. In particular, the mathematical formulation of the problem of interest is realized via a system of nonlinear functional equations (NFEs), and a rather general set of necessary and sufficient conditions for solvability is derived. The solution to the above system of NFEs can be proven to be a unique locally analytic one, and this enables the development of a series solution method that is easily programmable with the aid of a symbolic software package such as MAPLE. It is also shown that, on the basis of the solution to the above system of NFEs, a locally analytic manifold and a nonlinear control law can be explicitly derived that renders the manifold invariant for the class of skew-product systems considered. Furthermore, the restriction of the system dynamics on the aforementioned invariant manifold represents exactly the target controlled system dynamics. Finally, the proposed method is applied to the HF molecular system classically modeled as a rotationless Morse oscillator in the presence of an external laser-field, where the primary objective is molecular dissociation.     

Discrete morse theory and the cohomology ring

In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].