Characteristics of in-out intermittency in delay-coupled FitzHugh–Nagumo oscillators

We analyze a pair of delay-coupled FitzHugh–Nagumo oscillators exhibiting in-out intermittency as a part of the generating mechanism of extreme events. We study in detail the characteristics of in-out intermittency and identify the invariant subsets involved – a saddle fixed point and a saddle periodic orbit – neither of which are chaotic as in the previously reported cases of in-out intermittency. Based on the analysis of a periodic attractor possessing in-out dynamics, we can characterize the approach to the invariant synchronization manifold and the spiralling out to the saddle periodic orbit with subsequent ejection from the manifold. Due to the striking similarities, this analysis of in-out dynamics also explains in-out intermittency     

Construction of the Lyapunov Spectrum in a Chaotic System Displaying Phase Synchronization

We consider a three-dimensional chaotic system consisting of the suspension of Arnold’s cat map coupled with a clock via a weak dissipative interaction. We show that the coupled system displays a synchronization phenomenon, in the sense that the relative phase between the suspension flow and the clock locks to a special value, thus making the motion fall onto a lower dimensional attractor. More specifically, we construct the attractive invariant manifold, of dimension smaller than three, using a convergent perturbative expansion. Moreover, we compute via convergent series the Lyapunov exponents, including notably the central one. The result generalizes a previous construction of the attractive invariant manifold in a similar but simpler model. The main novelty of the current construction relies in the computation of the Lyapunov spectrum, which consists of non-trivial analytic exponents. Some conjectures about a possible smoothening transition of the attractor as the coupling is increased are also discussed.     

Cluster synchronization in networks of distinct groups of maps

In this paper, we study cluster synchronization in general bi-directed networks of nonidentical clusters, where all nodes in the same cluster share an identical map. Based on the transverse stability analysis, we present sufficient conditions for local cluster synchronization of networks. The conditions are composed of two factors: the common inter-cluster coupling, which ensures the existence of an invariant cluster synchronization manifold, and communication between each pair of nodes in the same cluster, which is necessary for chaos synchronization. Consequently, we propose a quantity to measure the cluster synchronizability for a network with respect to the given clusters via a function of the eigenvalues of the Laplacian corresponding to the generalized eigenspace transverse to the cluster synchronization manifold. Then, we discuss the clustering synchronous dynamics and cluster synchronizability for four artificial network models: (i) p-nearest-neighborhood graph; (ii) random clustering graph; (iii) bipartite random graph; (iv) degree-preferred growing clustering network. From these network models, we are to reveal how the intra-cluster and inter-cluster links affect the cluster synchronizability. By numerical examples, we find that for the first model, the cluster synchronizability regularly enhances with the increase of p, yet for the other three models, when the ratio of intra-cluster links and the inter-cluster links reaches certain quantity, the clustering synchronizability reaches maximal.     

Synchronization on coupled dynamical networks

In this paper, partial synchronization (PaS) in networks of coupled chaotic oscillator systems and synchronization in sparsely coupled spatiotemporal systems are explored. For the PaS, we reveal that the existence of PaS patterns depends on the symmetry property of the network topology, while the emergence of the PaS pattern depends crucially on the stability of the corresponding solution. An analytical criterion in judging the stability of PaS state on a given network are proposed in terms of a comparison between the Lyapunov exponent spectrum of the PaS manifold and that of the transversal manifold. The competition and selections of the PaS patterns induced by the presence of multiple topological symmetries of the network are studied in terms of the criterion. The phase diagram in distinguishing the synchronous and the asynchronous states is given. The criterion in judging PaS is further applied to the study of synchronization of two sparsely coupled spatiotemporal chaotic systems. Different synchronization regimes are distinguished. The present study reveals the intrinsic collective bifurcation of coupled dynamical systems prior to the emergence of global synchronization.     

In-out intermittency in partial differential equation and ordinary differential equation models

We find concrete evidence for a recently discovered form of intermittency, referred to as in-out intermittency, in both partial differential equation (PDE) and ordinary differential equation (ODE) models of mean field dynamos. This type of intermittency [introduced in P. Ashwin, E. Covas, and R. Tavakol, Nonlinearity 9, 563 (1999)] occurs in systems with invariant submanifolds and, as opposed to on-off intermittency which can also occur in skew product systems, it requires an absence of skew product structure. By this we mean that the dynamics on the attractor intermittent to the invariant manifold cannot be expressed simply as the dynamics on the invariant subspace forcing the transverse dynamics; the transverse dynamics will alter that tangential to the invariant subspace when one is far enough away from the invariant manifold. Since general systems with invariant submanifolds are not likely to have skew product structure, this type of behavior may be of physical relevance in a variety of dynamical settings. The models employed here to demonstrate in-out intermittency are axisymmetric mean-field dynamo models which are often used to study the observed large-scale magnetic variability in the Sun and solar-type stars. The occurrence of this type of intermittency in such models may be of interest in understanding some aspects of such variabilities. (c) 2001 American Institute of Physics.     

Invariant manifolds associated to nonresonant spectral subspaces

We show that, if the linearization of a map at a fixed point leaves invariant a spectral subspace which satisfies certain nonresonance conditions, the map leaves invariant a smooth manifold tangent to this subspace. This manifold is as smooth as the map—when the smoothness is measured in appropriate scales—but is unique amongC ^L invariant manifolds, whereL depends only on the spectrum of the linearization or on some more general smoothness classes that we detail. We show that if the nonresonance conditions are not satisfied, a smooth invariant manifold need not exist, and we also establish smooth dependence on parameters. We also discuss some applications of these invariant manifolds and briefly survey related work.     

Differential Forms and the Noncommutative Residue for Manifolds with Boundary in the Non-product Case

In this Letter, for an even-dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a four-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.     

Asymptotics of the Eta Invariant

We prove an asymptotic bound on the eta invariant of a family of coupled Dirac operators on an odd dimensional manifold. In the case when the manifold is the unit circle bundle of a positive line bundle over a complex manifold, we obtain precise formulas for the eta invariant.     

Sinusoidal synchronization of a Duffing oscillator with a chaotic pendulum

In this Letter, analytical conditions for the sinusoidal synchronization of the pendulum and Duffing oscillator are presented. From the analytical conditions, the invariant domain of such sinusoidal synchronization is determined, and the control parameter map of the synchronicity is achieved. Under specific parameters, numerical illustrations of the partial and full sinusoidal synchronizations of the controlled Duffing oscillator with the pendulum are carried out for a better understanding of such synchronization under specific function constraints. The methodology presented in this Letter is applicable to synchronizations with any specific function constraints.     

Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds

We examine synchronization of identical chaotic systems coupled in a drive/response manner. A rigorous criterion is presented which, if satisfied, guarantees that synchronization to the driving trajectory is linearly stable to perturbations. An easy to use approximate criterion for estimating linear stability is also presented. One major advantage of these criteria is that, for simple systems, many of the calculations needed to implement them can be performed analytically. Geometrical interpretations of the criterion are discussed, as well as how they may be used to investigate synchronization between mutual coupled systems and the stability of invariant manifolds within a dynamical system. Finally, the relationship between our criterion and results from control theory are discussed. Analytical and numerical results from tests of these criteria on four different dynamical systems are presented. (c) 1997 American Institute of Physics.     

Stalactite basin structure of dynamical systems with transient chaos in an invariant manifold

Dynamical systems with invariant manifolds occur in a variety of situations (e.g., identical coupled oscillators, and systems with a symmetry). We consider the case where there is both a nonchaotic attractor (e.g., a periodic orbit) and a nonattracting chaotic set (or chaotic repeller) in the invariant manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant manifold and another attractor not in the invariant manifold. It is found that the boundary separating these basins has an interesting structure: The basin of the attractor not in the invariant manifold is characterized by thin cusp shaped regions ("stalactites") extending down to touch the nonattracting chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these systems, and compare with numerical experiments. (c) 2000 American Institute of Physics.     

A remark on the rigidity of Poincaré–Einstein manifolds

In this paper, we give an optimal inequality relating the relative Yamabe invariant of a certain compactification of a conformally compact Poincaré–Einstein manifold with the Yamabe invariant of its boundary at infinity. As an application, we obtain an elementary proof (without any additional assumption) of the rigidity of the hyperbolic space as the only conformally compact Poincaré–Einstein manifold with the round sphere as its conformal infinity.

     

Comparison of Synchronization Ability of Four Types of Regular Coupled Networks

We investigate the synchronization ability of four types of regular coupled networks. By introducing the proper error variables and Lyapunov functions, we turn the stability of synchronization manifold into that of null solution of error equations, further, into the negative definiteness of some symmetric matrices, thus we get the sufficient synchronization stability conditions. To test the valid of the results, we take the Chua's circuit as an example. Although the theoretical synchronization thresholds appear to be very conservative, they provide new insights about the influence of topology and scale of networks on synchronization, and that the theoretical results and our numerical simulations are consistent.     

Global attractors for degenerate partial differential equations from nonlinear viscoelasticity

We study several problems for the forced motion of light, uniform, nonlinearly viscoelastic bodies carrying heavy attachments. A ‘reduced’ problem for such motions is obtained by setting the ratio of the inertia of the viscoelastic body to the inertia of the attachment equal to zero. Using methods from infinite-dimensional dynamical systems theory, we prove that the degenerate partial differential equation of this reduced problem has an attractor and that this attractor is contained in an invariant two-dimensional manifold on which solutions are governed by the classical ordinary differential equation for the forced motion of a particle on a massless spring.     

Comparison of Synchronization Ability of Four Types of Regular Coupled Networks

We investigate the synchronization ability of four types of regular coupled networks. By introducing the proper error variables and Lyapunov functions, we turn the stability of synchronization manifold into that of null solution of error equations, further, into the negative definiteness of some symmetric matrices, thus we get the sufficient synchronization stability conditions. To test the valid of the results, we take the Chua's circuit as an example. Although the theoretical synchronization thresholds appear to be very conservative, they provide new insights about the influence of topology and scale of networks on synchronization, and that the theoretical results and our numerical simulations are consistent.     

On a scale function for testing the conformality of a Finsler manifold to a Berwald manifold

In this paper, we are going to discuss the problem whether how we can check the conformality of a Finsler manifold to a Berwald manifold. The method is based on a differential 1-form constructing on the underlying manifold by the help of integral formulas such that its exterior derivative is conformally invariant. If the Finsler manifold is conformal to a Berwald manifold, then the exterior derivative vanishes. This gives the following necessary condition: the differential form is closed and, at least locally, it is exact as the exterior derivative of a scale function for testing the conformality. A necessary and sufficient condition is also given in terms of a distinguished linear connection on the underlying manifold – it is expressed by the help of canonical data. In order to illustrate how we can simplify the process in special cases Randers manifolds are considered with some explicit calculations.     

Adaptive exponential synchronization of delayed chaotic networks

This paper deals with the global exponential synchronization of a class of delayed chaotic networks. Under some simple conditions, the global synchronization of a network about its all variables is derived by only considering the global synchronization of its partial variables. Furthermore, based on the Halanay inequality technique, some delay-independent criteria are obtained to ensure the adaptive exponential synchronization of the model. And the simpler, less conservative and more efficient results are easy to be verified in engineering applications. Finally, an illustrative example is given to demonstrate the effectiveness of the presented synchronization scheme.     

Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

The extension of the ILDM concept to reaction–diffusion manifolds

In the present work, the method of simplifying chemical kinetics based on Intrinsic Low-Dimensional Manifolds (ILDMs) is modified to deal with the coupling of reaction and diffusion processes. Several problems of the ILDM method are overcome by a relaxation to an invariant system manifold (Reaction–Diffusion Manifold – REDIM). This relaxation process is governed by a multidimensional parabolic partial differential equation system, where, as an initial solution, an extended ILDM is used. Furthermore, a method for the solution and tabulation of the manifold is proposed in terms of generalized coordinates, with a subsequent procedure for the integration of the reduced system on the found manifold. This modification of the ILDM significantly improves the performance of the concept and allows us to extend its area of applicability. Illustrative comparative calculations of detailed and reduced models of flat laminar flames verify the approach.     

Phase transitions in geometrothermodynamics

Using the formalism of geometrothermodynamics, we investigate the geometric properties of the equilibrium manifold for diverse thermodynamic systems. Starting from Legendre invariant metrics of the phase manifold, we derive thermodynamic metrics for the equilibrium manifold whose curvature becomes singular at those points where phase transitions of first and second order occur. We conclude that the thermodynamic curvature of the equilibrium manifold, as defined in geometrothermodynamics, can be used as a measure of thermodynamic interaction in diverse systems with two and three thermodynamic degrees of freedom.