A general formalism for synchronization in finite dimensional dynamical systems

In this paper, an attempt is made to propose a general definition of synchronization for finite dimensional dynamical systems. The synchronization is defined here for two coupled dynamical systems with control inputs. Output functions of such systems are introduced to describe the systems’ properties on which the synchronization problem focus. Exact synchronization, asymptotic synchronization, and approximate synchronization are, respectively, defined by comparing the output functions in the corresponding ways. The definition here can also include chaos control and anti-control. The definition here covers various synchronization investigated in the references.     

Novikov-Morse theory for dynamical systems

The present paper contains an interpretation and generalization of Novikov's theory for Morse type inequalities for closed 1-forms in terms of concepts from Conley's theory for dynamical systems. We introduce the concept of a flow carrying a cocycle , (generalized) -flow for short, where is a continuous cocycle in bounded Alexander-Spanier cohomology theory. Gradient-like flows can then be characterized as flows carrying a trivial cocycle. We also define -Morse-Smale flows that allow the existence of “cycles” in contrast to the usual Morse-Smale flows. -flows without fixed points carry not only a cocycle, but a cohomology class, in the sense of [8], and we shall deduce a vanishing theorem for generalized Novikov numbers in that situation. By passing to a suitable cover of the underlying compact polyhedron adapted to the cocycle , we construct a so-called -Morse decomposition for an -flow. On this basis, we can use the Conley index to derive generalized Novikov-Morse inequalitites, extending those of M. Farber [12]. In particular, these inequalities include both the classical Morse type inequalities (corresponding to the case when is a coboundary) as well as the Novikov type inequalities ( when is a nontrivial cocycle). Received: 26 June 2001 / Accepted: 15 January 2002 / Published online: 6 August 2002     

A theory for flow switchability in discontinuous dynamical systems

The G-functions for discontinuous dynamical systems are introduced to investigate singularity in discontinuous dynamical systems. Based on the new G-function, the switchability of a flow from a domain to an adjacent one is discussed. Further, the full and half sink and source, non-passable flows to the separation boundary in discontinuous dynamical systems are discussed. A flow to the separation boundary in a discontinuous dynamical system can be passable or non-passable. Therefore, the switching bifurcations between the passable and non-passable flows are presented. Finally, the first integral quantity increment for discontinuous dynamical systems is given instead of the Melnikov function to develop the iterative mapping relations.     

Chaos synchronization between two different chaotic dynamical systems

This work presents chaos synchronization between two different chaotic systems by nonlinear control laws. First, synchronization problem between Genesio system and Rossler system has been investigated, and then the similar approach is applied to the synchronization problem between Genesio system and a new chaotic system developed recently in the literature. The control performances are verified by two numerical examples.     

Adaptive synchronization between two different chaotic dynamical systems

This work presents the synchronization between two different chaotic systems by using an adaptive feedback control scheme. The adaptive synchronization problem between an electrostatic system and electromechanical transducer has been investigated. An adaptive linear feedback law with two controllers is proposed to ensure the global chaos synchronization of the nonlinear electrostatic and electromechanical systems. Numerical simulations results are presented to demonstrate the effectiveness of the proposed method.     

A fast-slow dynamical systems theory for the Kuramoto type phase model

We present a fast-slow dynamical systems theory for the Kuramoto type phase model. When the order parameters are frozen, the fast system consists of independent oscillator equations, whereas the slow system describes the evolution of order parameters. We average out the slow system over the fast manifold to derive a weak form of an amplitude-angle coupled system for the evolution of Kuramoto?s order parameters. This yields the slow evolution of order parameters to be constant values which gives a rigorous proof to Kuramoto?s original assumption in his self-consistent mean-field theory.     

Ergodic theory of differentiable dynamical systems

Iff is a C^{1 + ɛ} diffeomorphism of a compact manifold M, we prove the existence of stable manifolds, almost everywhere with respect to everyf-invariant probability measure on M. These stable manifolds are smooth but do not in general constitute a continuous family. The proof of this stable manifold theorem (and similar results) is through the study of random matrix products (multiplicative ergodic theorem) and perturbation of such products. Dedicated to the memory of Rufus Bowen     

Deformation theory of quantizable dynamical systems

Deformations of families of quantizable dynamical systems are introduced and the local extension problem for quantizable dynamical systems is solved. This research was supported in part by NSF GP-13375 and NSF GP-20856. Entrata in Redazione il 2 dicembre 1970.     

Qualitative theory of p-adic dynamical systems

We consider a class of dynamical systems over the p-adic number field: hierarchical dynamical systems. We prove a strong variant of the Poincaré theorem on the number of returns for such systems and show that hierarchical systems do not admit mixing. We describe hierarchical dynamical systems over the projective line and present an example of a nonhierarchical p-adic system that admits mixing: the p-adic baker’s transformation.     

Generalized synchronization of two bidirectionally coupled discrete dynamical systems

Based on the modified system approach the generalized synchronization (GS) in two bidirectionally coupled discrete dynamical systems is classified into several types, and under some conditions, the existence, Lipschitz smoothness and Hölder continuity of two kinds of GS therein are derived and theoretically proved. In addition, numerical simulations validate the present theory.     

A framework for synchronization theory

A general framework is proposed for synchronization theory on finite dimensional dynamical systems with the intention to resolve the problem that puzzles some people of how to give a rigorous unified notion for describing the various synchronization phenomena in physical systems.     

Adaptive scheme for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems

In this paper, a new type of anticipating synchronization, called time-varying anticipating synchronization, is defined firstly. Then novel adaptive schemes for time-varying anticipating synchronization of certain or uncertain chaotic dynamical systems are designed based on the Lyapunov function and invariance principle. The update gain of coupling strength can be automatically adapted to a suitable strength depending on the initial values and can be properly chosen to adjust the speed of achieving synchronization, so these schemes are analytical and simple to implement in practice. A classical chaotic dynamical system is used to demonstrate the effectiveness of the proposed adaptive schemes with or without parameter uncertainties.     

Multivalued maps in stability theory of dynamical systems

Summary Multivalued maps like orbit, limit set, prolongations etc., are an useful tool in Dynamical Systems theory. In this work we develop a calculus for multivalued maps associated with a dynamical system. Then we give general definitions of stability and attraction of a compact set with respect to a multivalued map. On the basis of our calculus, we obtain several characterizations of stability and attraction, which generalise well known classical theorems. Such a general theory is applied to total stability of diffentiable dynamical systems. The equivalence among several approaches to total stability is established.     

Flows of stochastic dynamical systems: ergodic theory

We present a version of the Multiplicative Ergodic (Oseledec) Theorem for the flow of a nonlinear stochastic system definedon a smooth compact manifold. This theorem establishes the existence of a Lyapunov spectrum for the flow, which characterises the asymptotic behaviour of the derivative flow. Then we establish the existence of stable manifolds for the flow (on which trajectories cluster) associated with the Lyapunov spectrum. This work is a generalisation of that of Ruelle who deals with ordinary dynamical systems. Finally we give an example of a stochastic system for which the flow is calculated explicitly, and which illustrates the behaviour predicted by the abstract results.     

Contraction theory based adaptive synchronization of chaotic systems

Contraction theory based stability analysis exploits the incremental behavior of trajectories of a system with respect to each other. Application of contraction theory provides an alternative way for stability analysis of nonlinear systems. This paper considers the design of a control law for synchronization of certain class of chaotic systems based on backstepping technique. The controller is selected so as to make the error dynamics between the two systems contracting. Synchronization problem with and without uncertainty in system parameters is discussed and necessary stability proofs are worked out using contraction theory. Suitable adaptation laws for unknown parameters are proposed based on the contraction principle. The numerical simulations verify the synchronization of the chaotic systems. Also parameter estimates converge to their true values with the proposed adaptation laws.     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.