International Journal of Bifurcation and Chaos

We present and discuss certain global bifurcations involving the interaction of one- and two-dimensional invariant manifolds of steady and periodic solutions of the Kuramoto–Sivashinsky equation. Numerical bifurcation calculations, dimensionality reduction using approximate inertial manifolds/forms, as well as approximation and visualization of invariant manifolds are combined in order to characterize what we term the "Oseberg transition".

In this paper we show how to relate a form of high-dimensional complexity to chaotic and other types of dynamical systems. The derivation shows how "near-chaotic" complexity can arise without the presence of homoclinic tangles or positive Lyapunov exponents. The relationship we derive follows from the observation that the elements of invariant finite integer lattices of high-dimensional dynamical systems can, themselves, be viewed as single integers rather than coordinates of a point in n-space. From this observation it is possible to construct high-dimensional dynamical systems which have properties of shifts but for which there is no conventional topological conjugacy to a shift. The particular manner in which the shift appears in high-dimensional dynamical systems suggests that some forms of complexity arise from the presence of chaotic dynamics which are obscured by the large dimensionality of the system domain.

The effects of weak breaking of the midplane reflection symmetry on the 1:2 steady state mode interaction in Rayleigh–Bénard convection are discussed in a PDE setting. Effects of this type arise from the inclusion of non-Boussinesq terms or due to small differences in the boundary conditions at the top and bottom of the convecting layer. The latter provides the simplest realization, and captures all qualitative effects of such symmetry breaking. The analysis is performed for two Prandtl numbers, σ=10 and σ=0.1, representing behavior typical of large and low Prandtl numbers, respectively.

The variational inequality principle originated from a class of elliptic obstacle problem applied to the prediction of cavitation in the fluid film bearing and squeeze film damper, and the finite element method has been used to discretize the resulting elliptic variational inequality. Based on the Taylor series, the Jacobian matrix of film force with respect to location and velocity of journal and bearing house has been constructed, and it can be obtained from resolving a set of partial differential equations. An iteration algorithm based on the complementary property of the elliptic variational inequality has been also constructed to determine the rupture boundary, namely, free boundary.

On the basis of the fixed-interface eigen-mode and quasi-static modes, a method is constructed to condense the system. Furthermore, the Poincaré map is introduced to discretize the continuous flow, and the fixed point in the Poincaré section is the periodic solution and a method for calculating the Floquet multipliers is also constructed by resolving two sets of secondary ordinary differential equations in the form of matrix. The long-term dynamic behaviors and bifurcation of imbalance response of a rotor dynamic system, amounted in finite length fluid film bearings with squeeze film dampers in series, has been investigated based on the computation method mentioned above. In addition to the above, the influence of the truncation of mode series on the accuracy of solution in nonlinear dynamic system has been investigated tentatively by the comparison between mode series truncations, and the result obtained from the condensation method has been compared with the result obtained from the direct integration. All the results show that the method presented in this paper is effective in the prediction of cavitation in the fluid film and the bifurcation analysis of the system with fluid film.

We consider the dynamics of a two-degree of freedom impact oscillator subject to a motion limiting constraint. These systems exhibit a range of periodic and nonperiodic impact motions. For a particular set of parameters, we consider the bifurcations which occur between differing regimes of impacting motion and in particular those which occur due to a grazing bifurcation. Unexpected resonant behavior is also observed, due to the complexity of the dynamics. We consider both periodic and chaotic chatter motions and the regions of sticking which exist. Finally we consider the types of chaotic motion that occur within the parameter range. We discuss the possibility in relating successive low velocity impacts, especially with respect to possible low dimensional mappings for such a system.

Noninvertible circle maps may have a rotation interval instead of a unique rotation number. One may ask which of the numbers or sets of numbers within this rotation interval may be observed with positive probability in term of Lebesgue measure on the circle. We study this question numerically for families of circle maps. Both the interval and "observed" rotation numbers are computed for large numbers of initial conditions. The numerical evidence suggests that within the rotation interval only a very narrow band or even a unique rotation number is observed. These observed rotation numbers appear to be either locally constant or vary wildly as the parameter is changed. Closer examination reveals that intervals with wild variation contain many subintervals where the observed rotation numbers are locally constant. We discuss the formation of these intervals. We prove that such intervals occur whenever one of the endpoints of the rotation interval changes. We also examine the effects of various types of saddle-node bifurcations on the observed rotation numbers.

We study the dynamical systems given by generalized Chebyshev mappings and show that (1) the set of points with bounded orbits of F_c(z) is connected and its complement in C∪{∞} is simply connected if and only if -4 ≤ c ≤ 2; (2) if c > 2, then the set of points with bounded orbits of F_c(z) is Cantor set.

These results are the analogue of the theory of filled Julia sets of quadratic polynomials in one complex variable. We show that the mapping F_c(z) has relation to an important holomorphic map on the complex projective space P².

A one-dimensional chain consisting of many locally coupled oscillatory units is considered when both diffusive (linear) and nonlinear interactions exist. The dynamics of units may also be altered by a weak white Gaussian noise. Two different collective evolutionary paths have been identified in the chain: (i) the onset of temporal phase clusters and their subsequent breakdown or reconstruction through phase resetting, and (ii) the formation of a phase wave during a process of selforganization. Such a wave may originate from an initially disordered pattern after appropriate reorganization of phases and amplitudes occurring on a time lapse that can be very long.

We consider the implications of "all to all" coupling of n identical, Z₂ symmetric cells. In particular we demonstrate that how this coupling is achieved with respect to these internal symmetries is important, even in this very simple scenario. The coupling we consider leads to a full symmetry of S_n, Z₂×S_n or Z₂≀S_n, where the latter is the "wreath product" symmetry. By considering the generic solutions of each case on their respective irreducible representations, we compare and contrast the three cases, and show how very different solutions could arise in essentially similar systems.

The structure of the positive steady-state solutions for the Volterra–Lotka ecological model of two cooperating species is investigated. Some new results of existence, nonexistence and uniqueness are presented. The existence of a global attractor which is contained in an order interval is shown and the bounds of various dimensions of the attractor are given.

We study the asymptotic behavior of solutions for lattice dynamical systems. We first prove asymptotic compactness and then establish the existence of global attractors. The upper semicontinuity of the global attractor is also obtained when the lattice differential equations are approached by finite-dimensional systems.

Modular neural networks structured as associative memories are capable of processing inputs built from tensorial products of vectors. In this context, the operators of propositional and modal logic can be represented as modular distributed memories that can process not only classical Boolean but also fuzzy evaluations of truth-values of sentences. Furthermore, projecting memory outputs onto unit vectors yield discrete dynamical systems that exhibit varying degrees of complexity. As examples, we analyze outcomes of semantic evaluations in several self-referential systems including modal versions of the chaotic liar, antagonistic decisions and extended dilemmas. By studying these examples we hope to shed some light on the modeling of cognitive decisions.

This work investigates a class of lattice dynamical systems originated from cellular neural networks. In the vector field of this class, each component of the state vector and the output vector is related through a sigmoidal nonlinear output function. For two types of sigmoidal output functions, Liapunov functions have been constructed in the literatures. Complete stability has been studied for these systems using LaSalle's invariant principle on the Liapunov functions. The purpose of this presentation is two folds. The first one is to construct Liapunov functions for more general sigmoidal output functions. The second is to extend the interaction parameters into a more general class, using an approach by Fiedler and Gedeon. This presentation also emphasizes the complete stability when the equilibrium is not isolated for the standard cellular neural networks.

A contour detection based on a diffusive cellular nonlinear network is proposed. It is shown that there exists a particular nonlinear function for which, numerically, the obtained contour is satisfactory. Furthermore, this nonlinear function can be achieved using analog components.

This paper presents a decentralized linear feedback control for a lattice of coupled second-order systems with external forces. A classical linear PID control is used at each local site. A sufficient condition to obtain nonlocal results on stabilization and synchronization is provided. This condition does not depend on the system size neither on the coupling structure; hence, it allows us to obtain a systematic procedure for analyzing feedback control of coupled oscillators. Some implications of our results on locomotory motion control are discussed. Numerical examples are used to illustrate the theoretical results.

Concrete numerical examples of quadratic differential systems having three limit cycles surrounding one singular point are shown. In case another finite singular point also exists, a (3, 1) distribution of limit cycles is also obtained. This is the highest number of limit cycles known to occur in a quadratic differential system so far. Representative bifurcation diagrams are drawn for realistic parameter values.

The technique of using nonlinear approximations to increase the effectiveness of the OGY method of controlling chaotic dynamical systems is investigated and a way of further increasing the effectiveness of this general approach is presented.

This paper describes a simple method for calculating unstable periodic orbits (UPOs) and their control in piecewise-linear autonomous systems. The algorithm can be used to obtain any desired UPO embedded in a chaotic attractor, and the UPO can be stabilized by a simple state feedback control. A brief stability analysis of the controlled system is also given.

In this paper, we discuss the existence of strange nonchaotic attractors for the Chua's circuit with periodical excitation. We have studied the Lyapunov exponents, Poincaré maps, singular continuous spectra of characterizing the attractors. The results show that the excited Chua's circuit does indeed have the strange nonchaotic behaviors.

In this letter the so-called frequency switched Chua's circuit is presented. Its realization is based on the introduction of a switch device into the Chua's circuit, realized by using three State-Controlled CNN cells. The dynamics of this system has been investigated both numerically and experimentally with respect to the frequency of the switch enabling signal. It exhibits different strange attractors, according to the variation of the frequency of the switching signal. Moreover, very strange frequency hysteresis phenomena demonstrating the peculiarity of the introduced circuit have been revealed.

We present the structures and behaviors of cellular nonlinear networks (CNN) on spheres (spherical CNN for short). Although the cells in a spherical CNN can have any number of neighbors, in this paper we study the cases where each cell has only three neighbors. We present the simulation results of symmetric and asymmetric spherical CNNs. In our simulations, we use a Bucky-ball spherical CNN structure with 60 cells and an irregular spherical CNN structure with 100 cells. Since a spherical CNN has no boundary, we use an asymmetric spherical CNN to show how information propagates on spheres via local couplings. We also show that if a spherical CNN is locally active then different patterns can be found in its stable output.