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.     

A new approach for the computation of Hopf bifurcation points

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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.     

平面连续体多工况应力约束下结构拓扑优化的有无复合体方法

本文在[4]的基础上,按“有无复合体”模型对应力的约束下多工况平面连续体结构拓扑优化的求解进行研究,提出了“工况影响系数”的概念,用于“病态荷载”问题的识别的解决,获得了令人满意的数值结果。     

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.     

Metamorphoses of resonance curves in systems of coupled oscillators:The case of degenerate singular points

We study dynamics of two coupled periodically driven oscillators. The internal motion is separated off exactly to yield a nonlinear fourth-order equation describing inner dynamics. Periodic steady-state solutions of the fourth-order equation are determined within the Krylov–Bogoliubov–Mitropolsky approach and we compute the corresponding amplitude profiles.Metamorphoses of these amplitude curves induced by changes of control parameters as well as the corresponding changes of dynamics are studied within the framework of theory of differential properties of algebraic curves. The major finding is that there is a very rich dynamics in neighborhoods of degenerate singular points.     

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.     

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.     

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.     

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.