Cycle Chains and the LPA Model

The LPA model is a three dimensional system of nonlinear difference equations that has found many applications in population dynamics and ecology. In this paper, we consider a special case of the model (a case that approximates that used in experimental bifurcation and chaos studies) and prove several theorems concerning the existence and stability of periodic cycles, invariant loops, and chaos. Key is the notion of synchronous orbits (i.e. orbits lying on the coordinate planes). The main result concerns the existence of an invariant loop of synchronous orbits that bifurcates, in a nongeneric way, from the trivial equilibrium. The geometry and dynamics of this invariant loop are characterized. Specifically, it is shown that the loop is a cycle chain consisting of synchronous heteroclinic orbits connecting the three temporal phases of a synchronous 3-cycle. We also show that a period doubling route to chaos occurs within the class of synchronous orbits.     

Some Curious Phenomena in Coupled Cell Networks

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic 2n × 2n arrays of cells that generate 2ⁿ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

Stability and Bifurcation Analysis of a Nonlinear DDE Model for Drilling

We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by Stewart, Golubitsky, and Pivato. In particular we demonstrate patterns of synchrony in networks with small numbers of cells and in lattices (and periodic arrays) of cells that cannot readily be explained by conventional symmetry considerations. We also show that different types of dynamics can coexist robustly in single solutions of systems of coupled identical cells. The examples include a three-cell system exhibiting equilibria, periodic, and quasiperiodic states in different cells; periodic $2n\times 2n$ arrays of cells that generate $2^n$ different patterns of synchrony from one symmetry-generated solution; and systems exhibiting multirhythms (periodic solutions with rationally related periods in different cells). Our theoretical results include the observation that reduced equations on a center manifold of a skew product system inherit a skew product form.     

Coherence properties of cycling chaos

Cycling chaos is a heteroclinic connection between several chaotic attractors, at which switchings between the chaotic sets occur at growing time intervals. Here we characterize the coherence properties of these switchings, considering nearly periodic regimes that appear close to the cycling chaos due to imperfections or to instability. Using numerical simulations of coupled Lorenz, Roessler, and logistic map models, we show that the coherence is high in the case of imperfection (so that asymptotically the cycling chaos is very regular), while it is low close to instability of the cycling chaos.     

Effective approaches to explore rich dynamics of delay-differential equations

Discrete models are proposed to delve into the rich dynamics of nonlinear delayed systems under Euler discretization, such as backwards bifurcations, stable limit cycles, multiple limit-cycle bifurcations and chaotic behavior. The effect of breaking the special symmetry of the system is to create a wide complex operating conditions which would not otherwise be seen. These include multiple steady states, complex periodic oscillations, chaos by period doubling bifurcations. Effective computation of multiple bifurcations, stable limit cycles, symmetrical breaking bifurcations and chaotic behavior in nonlinear delayed equations is developed.     

Dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity

In this paper, we consider the dynamics and chaos control of the self-sustained electromechanical device with and without discontinuity. The amplitude equations are derived in the general case using the harmonic balance method. The model without discontinuity is first considered. The effects of the amplitude of the parametric modulation and some particular coefficients are found in the response curves. The transition to chaotic behavior is found using numerical simulations of the equations of motion. We find that chaos appears in the model between the quasi-periodic and periodic orbits when the amplitude of the external excitation E₀ vary. An adaptive Lyapunov control strategy enables us to drive the system from the chaotic states to a targeting periodic orbit. The effects of elasticity and damping on the dynamics of the self-sustained electromechanical system are also derived.     

Distributional chaos and irregular recurrence

For a continuous map φ:X→X of a compact metric space, we study relations between distributional chaos and the existence of a point which is quasi-weakly almost periodic, but not weakly almost periodic. We provide an example showing that the existence of such a point does not imply the strongest version of distributional chaos, DC1. Using this we prove that, even in the class of triangular maps of the square, there are no relations to DC1. This result, among others, contributes to the solution of a problem formulated by A.N. Sharkovsky in the eighties.     

Analytic invariant manifolds for sequences of diffeomorphisms

We obtain real analytic invariant manifolds for trajectories of maps assuming only the existence of a nonuniform exponential behavior. We also consider the more general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We emphasize that the maps that we consider are defined in a real Euclidean space, and thus, one is not able to obtain the invariant manifolds from a corresponding procedure to that in the nonuniform hyperbolicity theory in the context of holomorphic dynamics. We establish the existence both of stable (and unstable) manifolds and of center manifolds. As a byproduct of our approach we obtain an exponential control not only for the trajectories on the invariant manifolds, but also for all their derivatives.     

Game-theoretic coupled riccati equations associated to controlled linear differential systems with jump markov perturbations

In this paper we investigate several properties of the stabilizing solution of a class of systems of Riccati type differential equations with indefinite sign associated to controlled systems described by differential equations with Markovian jumping.

We show that the existence of a bounded on R ₊ and stabilizing solution for this class of systems of Riccati type differential equations is equivalent to the solvability of a control-theoretic problem, namely disturbance attenuation problem.

If the coefficients of the considered system are theta;-periodic functions then the stabilizing solution is also theta;-periodic and if the coefficients are asymptotic almost periodic functions, then the stabilizing solution is also asymptotic almost periodic and its almost periodic component is a stabilizing solution for a system of Riccati type differential equations defined on the whole real axis. One proves also that the existence of a stabilizing and bounded on R ₊ solution of a system of Riccati differential equations with indefinite sign is equivalent to the existence of a solution to a corresponding system of matrix inequalities. Finally, a minimality property of the stabilizing solution is derived.     

Some results on the large-time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations

Systems of Hamilton–Jacobi equations arise naturally when we study optimal control problems with pathwise deterministic trajectories with random switching. In this work, we are interested in the large-time behavior of weakly coupled systems of first-order Hamilton–Jacobi equations in the periodic setting. First results have been obtained by Camilli et al. (NoDEA Nonlinear Diff Eq Appl, 2012) and Mitake and Tran (Asymptot Anal, 2012) under quite strict conditions. Here, we use a PDE approach to extend the convergence result proved by Barles and Souganidis (SIAM J Math Anal 31(4):925–939 (electronic), 2000) in the scalar case. This result permits us to treat general cases, for instance, systems of nonconvex Hamiltonians and systems of strictly convex Hamiltonians. We also obtain some other convergence results under different assumptions. These results give a clearer view on the large-time behavior for systems of Hamilton–Jacobi equations.     

Pseudo-stable and Pseudo-unstable Fiber Bundles for Dynamic Equations on Measure Chains

Invariant fiber bundles are the generalization of invariant manifolds from classical discrete or continuous dynamical systems to non-autonomous dynamic equations on measure chains. In this paper, we present a self-contained proof of their existence and smoothness. Our main result generalizes the so-called "Hadamard-Perron-Theorem" for hyperbolic finite-dimensional diffeomorphisms to pseudo-hyperbolic time-dependent non-regressive dynamic equations in Banach spaces. The proof of their smoothness uses a fixed point theorem of Vanderbauwhede-Van Gils.     

Reciprocal inhibitory coupling: Measure and control of chaos on a biophysically motivated model of bursting

Bursting activity is an interesting feature of the temporal organization in many cell firing patterns. This complex behavior is characterized by clusters of spikes (action potentials) interspersed with phases of quiescence. As shown in experimental recordings, concerning the electrical activity of real neurons, the analysis of bursting models reveals not only patterned periodic activity but also irregular behavior [1], [2]. The interpretation of experimental results, particularly the study of the influence of coupling on chaotic bursting oscillations, is of great interest from physiological and physical perspectives. The inability to predict the behavior of dynamical systems in presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we focus our attention on a specific class of biophysically motivated maps, proposed in the literature to describe the chaotic activity of spiking–bursting cells [Cazelles B, Courbage M, Rabinovich M. Anti-phase regularization of coupled chaotic maps modelling bursting neurons. Europhys Lett 2001;56:504–9]. More precisely, we study a map that reproduces the behavior of a single cell and a map used to examine the role of reciprocal inhibitory coupling, specially on two symmetrically coupled bursting neurons. Firstly, using results of symbolic dynamics, we characterize the topological entropy associated to the maps, which allows us to quantify and to distinguish different chaotic regimes. In particular, we exhibit numerical results about the effect of the coupling strength on the variation of the topological entropy. Finally, we show that complicated behavior arising from the chaotic coupled maps can be controlled, without changing of its original properties, and turned into a desired attracting time periodic motion (a regular cycle). The control is illustrated by an application of a feedback control technique developed by Romeiras et al. [Romeiras FJ, Grebogi C, Ott E, Dayawansa WP. Controlling chaotic dynamical systems. Physica D 1992;58:165–92]. This work provides an illustration of how our understanding of chaotic bursting models can be enhanced by the theory of dynamical systems.     

Analysis of an epidemic model with density-dependent birth rate,birth pulses

In this paper, we propose an SIS epidemic model for which population births occur during a single period of the year. Using the discrete map, we obtain exact periodic solutions of system which is with Ricker function. The existence and stability of the infection-free periodic solution and the positive periodic solution are investigated. The Poincaré map, the center manifold theorem and the bifurcation theorem are used to discuss flip bifurcation and bifurcation of the positive periodic solution. Numerical results imply that the dynamical behaviors of the epidemic model with birth pulses are very complex, including small-amplitude periodic 1 solution, large-amplitude multi-periodic cycles, and chaos. This suggests that birth pulse, in effect, provides a natural period or cyclicity that allow for a period-doubling route to chaos.     

New approach to the analysis of Hamiltonian and conservative systems

We suggest a new approach to the analysis of solutions of complicated conservative (in particular, Hamiltonian) systems, which implies the construction of an approximating extended two-parameter dissipative system of equations whose stable solutions (attractors) are arbitrarily exact approximations to solutions of the original conservative system. On the basis of numerical experiments for several conservative and Hamiltonian systems with two degrees of freedom, we show that, in all these systems, transition to chaos takes place not through the destruction of two-dimensional tori of the unperturbed system but, conversely, through the generation of complicated two-dimensional tori around cycles of the extended dissipative system and through an infinite cascade of bifurcations of the generation of new cycles and singular trajectories in accordance with the Feigenbaum-Sharkovskii-Magnitskii theory.     

Chaos anticontrol and synchronization of three time scales brushless DC motor system

Chaos anticontrol of three time scale brushless dc motors and chaos synchronization of different order systems are studied. Nondimensional dynamic equations of three time scale brushless DC motor system are presented. Using numerical results, such as phase diagram, bifurcation diagram, and Lyapunov exponent, periodic and chaotic motions can be observed. By adding constant term, periodic square wave, the periodic triangle wave, the periodic sawtooth wave, and kx|x| term, to achieve anticontrol of chaotic or periodic systems, it is found that more chaotic phenomena of the system can be observed. Then, by coupled terms and linearization of error dynamics, we obtain the partial synchronization of two different order systems, i.e. brushless DC motor system and rate gyroscope system.     

Periodic and homoclinic solutions generated by impulses for asymptotically linear and sublinear Hamiltonian system

In this paper, we get the existence of periodic and homoclinic solutions for a class of asymptotically linear or sublinear Hamiltonian systems with impulsive conditions via variational methods. However, without impulses, there is no homoclinic or periodic solution for the system considered in this paper. Moreover, our results can be used to study the existence of periodic and homoclinic solutions of difference equations.     

Unboundedness and periodicity of solutions for a discrete-time network model of three neurons

In this work, we consider a system of delay difference equations with McCulloch-Pitts nonlinearity, which describes the dynamics of a network of three neurons with no internal decays. It is shown that every solution is either truncated periodic or unbounded. Our results have potential applications in neural networks.     

多频激励软弹簧型Duffing系统中的混沌

研究了多频激励下的软弹簧型Duffing系统的混沌动力学,发现混沌产生的根本原因是系统相空间中横截异宿环面的存在．建立了双频激励情况下二维环面上的Poincaré映射、稳定流形和不稳定流形,应用Melnikov方法给出了稳定流形和不稳定流形横截相交的条件,并将此方法推广到激励包含有限多个频率的情形．推广了Melnikov方法在高维系统中的应用,给出了Smale马蹄意义下混沌存在的判据．同时证明,激励频率数目的增加扩大了参数空间上的混沌区域．     

Resonance bifurcation from homoclinic cycles

Differential equations that are equivariant under the action of a finite group can possess robust homoclinic cycles that can moreover be asymptotically stable. For differential equations in R⁴ there exists a classification of different robust homoclinic cycles for which moreover eigenvalue conditions for asymptotic stability are known. We study resonance bifurcations that destroy the asymptotic stability of robust ‘simple homoclinic cycles’ in four-dimensional differential equations. We establish that typically a periodic trajectory near the cycle is created, asymptotically stable in the supercritical case.