Simplified predictive criteria for the onset of chaos

Two alternative criteria for predicting the onset of chaos are presented. Both are based on the notion that it is the interaction between a stable and nearby unstable limit cycle pair in the phase space that disrupts the stable motion, thereby producing chaotic behavior. The first criterion is based upon an intersection of the unstable and stable limit cycle orbits in the phase plane. The second criterion proposes that an energy equivalence between the stable and unstable limitcycles may be responsible for the loss of periodicity of the stable motion. Both criteria are tested numerically using three distinct softening spring oscillators and their predictive capabilities are discussed. The results of this study, particularly for the energy criterion, are encouraging.     

Chaotic mixing and mixing efficiency in a short time

Several studies of the chaotic motion of fluid particles by two-dimensional time-periodic flows or three-dimensional steady flows, called Lagrangian chaos, are first introduced. Secondly, some of the studies on efficient mixing caused by Lagrangian chaos, called chaotic mixing, are reviewed with discussion of several indices for the estimation of mixing efficiency. Finally, several indices to estimate the efficiency of mixing in a short time, such as those related to transport matrices, stable and unstable manifolds of hyperbolic periodic points of Poincaré maps, and lines of separation, are explained by showing examples of mixing by two-dimensional time-periodic flows between eccentric rotating cylinders and mixing by three-dimensional steady flows in a model of static mixers.     

On homoclinic bifurcation system in the presence of stochastic perturbation

By virture of the singular point theory for one-dimension diffusion process and the stochastic averaging approach of energy envelop, the bifurcation behavior of a homoclinic bifurcation system, which is in the presence of parametric white noise and is concealed behind a codimension two bifurcation point, is investigated in this paper. Supported by the National Science Foundation of China under Grant No. 19602016.     

Singular perturbation theory for homoclinic orbits in a class of near-integrable Hamiltonian systems

This paper describes a new type of orbits homoclinic to resonance bands in a class of near-integrable Hamiltonian systems. It presents a constructive method for establishing whether small conservative perturbations of a family of heteroclinic orbits that connect pairs of points on a circle of equilibria will yield transverse homoclinic connections between periodic orbits in the resonance band resulting from the perturbation. In any given example, this method may be used to prove the existence of such transverse homoclinic orbits, as well as to determine their precise shape, their asymptotic behavior, and their possible bifurcations. The method is a combination of the Melnikov method and geometric singular perturbation theory for ordinary differential equations.     

Computational methods for global analysis of homoclinic and heteroclinic orbits: A case study

In earlier paper we have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ⁿ . The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. In this paper we extend our algorithm to incorporate higher-order approximations of the unstable and stable manifolds. This approximation is especially useful if we want to compute center manifolds accurately. A procedure for switching between the periodic approximation of homoclinic orbits and the higher-order approximation of homoclinic orbits provides additional flexibility to the method. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.     

Continuation of limit cycles near saddle homoclinic points using splines in phase space

We present a new algorithm for continuation of limit cycles of autonomous systems as a system parameter is varied. The algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. Currently popular algorithms in bifurcation analysis packages compute time-domain approximations of limit cycles using either shooting or collocation. The present approach seems useful for continuation near saddle homoclinic points, where it encounters a corner while time-domain methods essentially encounter a discontinuity (a relatively short period of rapid variation). Other phase space-based algorithms use rescaled arclength in place of time, but subsequently resemble the time-domain methods. Compared to these, we introduce additional freedom through a variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented. Comparisons with results from the popular package, MATCONT, are favorable close to saddle homoclinic points.     

Chaos in the Convective Flow of a Fluid Mixture in a Porous Medium

The results of the study of the global behaviour of the convective flow of a binary mixture in a porous medium are presented. Bifurcation diagram, fixed points, periodic, chaotic solutions, stable and unstable manifolds, and basins of attraction have been calculated. Different behaviours (chaos, undecidable behaviour, etc.) have been found.     

A note on the critical points of equilibrium paths in imperfect structures

In a recent work (Int. J. Solids Struct. 37 (2000) 1561) by one of the authors, an extended system for calculating critical points of equilibrium paths in imperfect structures was presented. However, the extremum nature of these points was not analyzed explicitly in that paper. In this note, we will fill in the gap and establish a sufficient condition for determining the buckling strength of imperfect structures.     

On the Chaotic Instability of a Nonsliding Liquid-Filled Top with a Small Spheroidal Base via Melnikov-Holmes-Marsden Integrals

Chaotic orientations of a top containing a fluid filled cavity are investigated analytically and numerically under small perturbations. The top spins and rolls in nonsliding contact with a rough horizontal plane and the fluid in the ellipsoidal shaped cavity is considered to be ideal and describable by finite degrees of freedom. A Hamiltonian structure is established to facilitate the application of Melnikov-Holmes-Marsden (MHM) integrals. In particular, chaotic motion of the liquid-filled top is identified to be arisen from the transversal intersections between the stable and unstable manifolds of an approximated, disturbed flow of the liquid-filled top via the MHM integrals. The developed analytical criteria are crosschecked with numerical simulations via the 4th Runge-Kutta algorithms with adaptive time steps.     

Multi-modes control method for spacecraft formation establishment and reconfiguration near the libration points

This paper studies Multi-modes control method for libration points formation establishment and reconfiguration. Firstly, relations between optimal impulse control and Floquet modes are investigated. Method of generating modes is proposed. Characteristics of the mode coefficients stimulated at different time are also given. Studies show that coefficients of controlled modes can be classified into four types, and formation establishment and reconfiguration can be achieved by multi-impulse control with the presented method of generating modes. Then, since libration points formation is generally unstable, mutli-modes keeping control method which can stabilize five Floquet modes simultaneously is proposed. Finally, simulation on formation establishment and reconfiguration are carried out by using method of generating modes and mutli-modes keeping control method. Results show that the proposed control method is effective and practical.     

Stability of triangular libration points in a planar restricted elliptic three body problem in cases of double resonances

Stability of triangular libration points in a planar restricted elliptic three-body problem is investigated for two sets of parameters corresponding to the cases of double resonances. These positions are shown to be formally stable in the case of the fourth-order double resonance. When a weak third-order and a strong fourth-order resonances occur in the system, any motion of an approximate autonomous system is found to be bounded if initial conditions are sufficiently close to the unperturbed motion. Boundedness domain estimate is obtained.     

Global bifurcations in the motion of parametrically excited thin plates

In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovai and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.     

Computational recovery of the homoclinic orbit in porous media convection

A simple procedure for computational recovery of the homoclinic orbit in porous media convection is presented.     

Temporal Dynamics of Brain Activity in Human Memory Processes

We propose a line of study by which Functional Magnetic Resonance Imaging (FMRI) can be used together with nonlinear dynamics concepts as a medium for the study of brain organization. The concentration is on (a) the complex behavior of elementary neural circuits, and how they interact over brief spans of time to produce cognition and memory; and (b) the change in circuit patterns associated with aging. The method of orbital decomposition appears to be ideally suited for these objectives and for determining how they integrate into hierarchical processes. The adapted procedure begins with a 3-D FMRI matrix of metabolic activity. Recurring patterns within a matrix row are identified and matched across rows and across depth slices. These hierarchical patterns are then compared over time for further recurrences. The computational procedure identifies the optimal pattern length over time, the patterns, and the largest Lyapunov for the system of patterns. Computations are assisted by statistical tests for the extent to which the isolated patterns represent the underlying data.     

Symbolic Dynamic Patterns of Written Exchanges: Hierarchical Structures in an Electronic Problem Solving Group

A spontaneous scientific creative problem solving discussion, which had transpired across a listserver, was captured and content-analyzed according to two protocols. One protocol centered on social interaction and the second pertained to problem completion behavior. Data were analyzed for topological entropy and dimensional complexity with a hierarchical variation of the symbolic dynamics procedure that was first introduced by Guastello, Hyde, and Odak (1998). The principal findings were that (1) There was a greater prevalence of clarification and initiating (particularly creative) behaviors compared to a real-time creative problem solving episode; they were presumably liberated by the unblocking effect that tends to characterize computer-assisted group discussions. (2) There were complex responses containing up to five conversational elements as identified by a Social Interaction scoring system. (3) The two different scoring protocols produced different indicators of string length and entropy. The content of those patterns, which are, in principle, decomposed orbits, is discussed.     

Control of the homoclinic bifurcation in buckled beams: Infinite dimensional vs reduced order modeling

A method for controlling non-linear dynamics and chaos is applied to the infinite dimensional dynamics of a buckled beam subjected to a generic space varying time-periodic transversal excitation. The homoclinic bifurcation of the hilltop saddle is identified as the undesired dynamical event, because it triggers, e.g., cross-well scattered (possibly chaotic) dynamics. Its elimination is then pursued by a control strategy which consists in choosing the best spatial and temporal shape of the excitation permitting the maximum shift of the homoclinic bifurcation threshold in the excitation amplitude-frequency parameters space.The homoclinic bifurcation is detected by the Holmes and Marsden's theorem [A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam, Arch. Ration. Mech. Anal. 76 (1981) 135-165] constituting a generalization of the classical Melnikov's theory. Two classes of boundary conditions (b.c.) are identified: for the first, the Melnikov function is exactly the same as obtained with the reduced order models, while for the second, which is more general, this is no longer true, and the non-linear normal modes theory is used. Based on this distinction, the control method is then separately applied to the two cases, and the optimal spatial and temporal shapes of the excitation are determined.A detailed comparison of the infinite vs finite dimensional models is performed with respect to the control features, and it is shown that, depending on the b.c., the control based on the reduced order model provides either exact or engineering acceptable results, although more systematic investigations are required to generalize the last conclusion.     

Numerical analysis and accurate computation of heteroclinic orbits in the case of center manifolds

In earlier papers (see the preceding paper and the references there), Doedel and the author have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ⁿ in the case that the solution approaches the fixed points exponentially. The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. Using the fact that the linearized operator of the problem is Fredholm in Banach spaces with exponential weights, the authors employed the general theory of approximation of nonlinear problems to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. In this paper we extend the analysis in the preceding paper to the case of center manifolds which requires the refinement of the analysis in the preceding paper. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.     

On the Global Geometric Structure of the Dynamics of the Elastic Pendulum

We approach the planar elastic pendulum as a singular perturbation of the pendulum to show that its dynamics are governed by global two-dimensional invariant manifolds of motion. One of the manifolds is nonlinear and carries purely slow periodic oscillations. The other one, on the other hand, is linear and carries purely fast radial oscillations. For sufficiently small coupling between the angular and radial degrees of freedom, both manifolds are global and orbitally stable up to energy levels exceeding that of the unstable equilibrium of the system. For fixed value of coupling, the fast invariant manifold bifurcates transversely to create unstable radial oscillations exhibiting energy transfer. Poincaré sections of iso-energetic manifolds reveal that only motions on and near a separatrix emanating from the unstable region of the fast invariant manifold exhibit energy transfer.