Noise-induced chaotic motions in Hamiltonian systems with slow-varying parameters

This paper studies chaotic motions in quasi-integrable Hamiltonian systems with slow-varying parameters under both harmonic and noise excitations. Based on the dynamic theory and some assumptions of excited noises, an extended form of the stochastic Melnikov method is presented. Using this extended method, the homoclinic bifurcations and chaotic behavior of a nonlinear Hamiltonian system with weak feed-back control under both harmonic and Gaussian white noise excitations are analyzed in detail. It is shown that the addition of stochastic excitations can make the parameter threshold value for the occurrence of chaotic motions vary in a wider region. Therefore, chaotic motions may arise easily in the system. By the Monte-Carlo method, the numerical results for the time-history and the maximum Lyapunov exponents of an example system are finally given to illustrate that the presented method is effective.     

Noise-induced bistable switching dynamics through a potential energy landscape

Interlinked positive feedback loops,an important building block of biochemical systems,can induce bistable switching,leading to long-lasting state changes by brief stimuli.In this work,prevalent mutual activation between two species as another positive feedback is added to a generic interlinked positive-feedback-loop model originating from many realistic biological circuits.A stochastic fluctuation of the positive feedback strength is introduced in a bistable interval of the feedback strength,and bistability appears for the moderate feedback strength at a certain noise level.Stability analysis based on the potential energy landscape is further utilized to explore the noise-induced switching behavior of two stable steady states.     

Noise-induced outer synchronization between two different complex dynamical networks

In this paper, based on the theory of stochastic differential equations, we study the outer synchronization between two different complex dynamical networks with noise coupling. The theoretical result shows that two different complex networks can achieve generalized outer synchronization only with white-noise-based coupling. Numerical examples further verify the effectiveness and feasibility of the theoretical results. Numerical evidence shows that the synchronization rate is proportional to the noise intensity.     

Noise-induced chaos in the elastic forced oscillators with real-power damping force

The chaotic behavior of the elastic forced oscillators with real-power exponents of damping and restoring force terms under bounded noise is investigated. By using random Melnikov method, a mean square criterion is used to detect the necessary conditions for chaotic motion of this stochastic system. The results show that the threshold of bounded noise amplitude for the onset of chaos in the system increases as the intensity of the random frequency increases, and decrease as the real-power exponent of damping term increase. The threshold of bounded noise amplitude for the onset of chaos is determined by the numerical calculation via the largest Lyapunov exponents. The effects of bounded noise and real-power exponent of damping term on bifurcation and Poincaré map are also investigated. Our results may provide a valuable guidance for understanding the effect of bounded noise on a class of generalized double well system.     

Noise-induced phenomena in a versatile class of prototype dynamical system with time delay

Nonlinear Dynamics - This work presents a general class of prototype birhythmic dynamical systems, which can be extensively used to study the generation of complex bifurcation of limit cycles....     

Efficient potential well escape for bi-stable Duffing oscillators

The problem of escape from a potential well of bi-stable oscillators has attracted attention given the diversity of physical and engineering systems described by this mathematical model. Most previous studies have considered quasi-static dynamics leading to escape. In devising efficient escape strategies for structures, transient conditions have not yet received adequate consideration. In this study, the intra-well nonlinear resonant dynamics of bi-stable systems are studied and exploited, yielding a time-efficient strategy for triggering minimal amplitude escape by employing transient perturbations. The response characteristics of both, the symmetric and asymmetric double-well Duffing oscillators are explored analytically to identify the stable solution branches for any given forcing configuration. Based on the basins of attraction of the stable attractors, a novel actuation methodology employing controlled perturbations in the phase of the forcing for driving the system into a series of high-amplitude limit cycle oscillations and eventual escape to the desired stable solution is proposed. Additionally, accelerated settling to the desired configuration is achieved by implementing state feedback techniques. The proposed algorithm serves as a potential tool for implementing fast shape adaptation in bi-stable structural systems.     

Cross-well chaos and escape phenomena in driven oscillators

The paper is devoted to the study of common features in regular and strange behavior of the three classic dissipative softening type driven oscillators: (a) twin-well potential system, (b) single-well potential unsymmetric system and (c) single-well potential symmetric system.Computer simulations are followed by analytical approximations. It is shown that the mathematical techniques and physical concepts related to the theory of nonlinear oscillations are very useful in predicting bifurcations from regular, periodic responses to cross-well chaotic motions or to escape phenomena. The approximate analysis of periodic, resonant solutions and of period doubling or symmetry breaking instabilities in the Hill's type variational equation provides us with closed-form algebraic simple formulae; that is, the relationship between critical system parameter values, for which strange phenomena can be expected.     

Potential well escape in a galloping twin-well oscillator

Nonlinear Dynamics - When a bi-stable oscillator undergoes a supercritical Hopf bifurcation due to a galloping instability, intra-well limit cycle oscillations of small amplitude are born. The...     

Asteroid debris: Temporary capture and escape orbits

We investigate the dynamical behaviour of debris ejected from the surface of an asteroid, due to a generic – natural or artificial – surface process. We make an extensive statistical study of the dynamics of particles flowing from the asteroid. We observe different behaviours: particles which fall again on the asteroid surface, or rather escape from its gravitational field or are temporary trapped in orbit around the asteroid. The tests are made by varying different parameters, like the size of the asteroid, its eccentricity, the angular velocity of the asteroid, the area-to-mass ratio of the debris.We also extend the study to the case of a sample of binary asteroids with a mass ratio equal to 10⁻³; we vary the distance of the moonlet from the asteroid, to see its effect on the debris dynamics.Our simulations aim to identify regions where the debris can temporarily orbit around the asteroid or rather escape from it or fall back on the surface. These results give an important information on where a spacecraft could be safely stay after the end of the process which has produced the debris.     

Noise-induced snap-through of a buckled column with continuously distributed mass: A chaotic dynamics approach

For a spatially-extended dynamical system we illustrate the use of a chaotic dynamics approach to obtain criteria on the occurrence of noise-induced escapes from a preferred region of phase space. Our system is a buckled column with continuous mass, subjected to a transverse continuously distributed load that varies randomly with time. We obtain a stochastic counterpart of the Melnikov necessary condition for chaos—and snap-through—derived by Holmes and Mardsen for the harmonic loading case. Our approach yields a lower bound for the probability that snap-through cannot occur during a specified time interval. In particular, for excitations with finite-tailed marginal distribution, a simple criterion is obtained that guarantees the non-occurrence of snap-through.     

Transition to escape in a system of coupled oscillators

We study a Hamiltonian system of coupled oscillators derived from two forced pendulums, connected with a torsional spring. The uncoupled limit is described by two identical oscillators, each possessing a homoclinic orbit separating bounded from unbounded motion. We focus on intermediate energy levels which lead to detained motions, defined as trajectories that, though unbounded as t → ∞, oscillate within the region defined by the homoclinic orbit of the unperturbed system for a long but finite time. We analyze the existence and behavior of these motions in terms of equipotential surfaces. These curves provide bounds on the motion of the system and are shown to be closed for low energies. However, above some critical energy level the equipotential curves become open. The detained trajectories are shown to arise from the region of phase space that was, for appropriate energies, stochastic. These motions remain within this region for long times before finally “leaking out” of the opening in the equipotential curves and proceeding to infinity.     

Elucidating the escape dynamics of the four hill potential

This paper treads discontinuous bifurcation in piecewise smooth systems of Filippov type. These bifurcations occur when a fixed point or a periodic orbit crosses with the border between two regions of smooth behavior. A detailed analysis of generalization Poincaré map and monodromy matrix which are related shows that subfamily of system with invariant cone-like objects is foliated by periodic orbits and determines its stability. In addition, we introduce a theoretical framework for analyzing 3D perturbed nonlinear piecewise smooth systems and give necessary conditions so that different types of bifurcations occur. The analysis identifies criteria for the existence of a novel bifurcation based on sensitively the location of the return map. Moreover, the piecewise smooth Melnikov function and sufficient conditions of the existence of the periodic orbits for nonlinear perturbed system are explicitly obtained.     

Side escape of fluid from a stream through a nozzle

The coefficient of compression during escape from nozzles at a 60 angle to the fluid stream is found for the case of irrotational motion in the stream and the nozzle. Conditions are determined under which such a stream will exist. It is shown that an increase in the relative length of the nozzle does not affect the value of the compression coefficient.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 156–159, March–April, 1975.     

Expected escape times from attractor basins due to low intensity noise

In this paper, a numerical approach is described to estimate escape times from attractor basins when a dynamical system is subjected to noise or stochastic perturbations. Noise can affect nonlinear system response by driving solution trajectories to different attractors. The changes in physical behavior can be observed as amplitude and phase change of periodic oscillations, initiation or annihilation of chaotic motion, phase synchronization, and so on. Estimating probability of transitions from one attractor to another, and predicting escape times are essential for quantifying the effects of noise on the system response. In this paper, a numerical approach is outlined where probability transition maps are generated between grids. Then, these maps are iterated to find the probability distribution after long durations, wherein, a constant escape rate can be observed between basins. The constant escape rate is then used to estimate the average escape times. The approach is applicable to systems subjected to low-intensity stochastic disturbances and with long escape times, where Monte Carlo simulations are impractical. Escape times up to \(10^{13}\) periods are estimated without relying on computationally expensive computations.

     

Analytic exploration of safe basins in a benchmark problem of forced escape

Nonlinear Dynamics - The paper presents an analytical approach to predicting the safe basins (SBs) in a plane of initial conditions (ICs) for escape of classical particle from...     

Trapping and escape of dislocations in micro-crystals with external and internal barriers

We perform three-dimensional dislocation dynamics simulations of solid and annular pillars, having both free-surface boundary conditions, or strong barriers at the outer and/or inner surfaces. Both pillar geometries are observed to exhibit a size effect where smaller pillars are stronger. The scaling observed is consistent with the weakest-link activation mechanism and depends on the solid pillar diameter, or the annular pillar effective diameter, D_eff = D − D_i, where D and D_i are the external and internal diameters of the pillar, respectively. An external strong barrier is observed to dramatically increase the dislocation density by an order of magnitude due to trapping dislocations at the surface. In addition, a considerable increase in the flow strength, by up to 60%, is observed compared to simulations having free-surface boundary conditions. As the applied load increases, weak spots form on the surface of the pillar by dislocations breaking through the surface when the RSS is greater than the barrier strength. The hardening rate is also observed to increase with increasing barrier strength. With cross-slip, we observe dislocations moving to other glide planes, and sometimes double-cross-slipping, producing a thickening of the slip traces at the surface. Finally the results are in qualitative agreement with recent compression experimental results of coated and centrally-filled micropillars.     

The route to chaos for the Kuramoto-Sivashinsky equation

We present the results of extensive numerical experiments of the spatially periodic initial value problem for the Kuramoto-Sivashinsky equation. Our concern is with the asymptotic nonlinear dynamics as the dissipation parameter decreases and spatio-temporal chaos sets in. To this end the initial condition is taken to be the same for all numerical experiments (a single sine wave is used) and the large time evolution of the system is followed numerically. Numerous computations were performed to establish the existence of windows, in parameter space, in which the solution has the following characteristics as the viscosity is decreased: a steady fully modal attractor to a steady bimodal attractor to another steady fully modal attractor to a steady trimodal attractor to a periodic (in time) attractor, to another steady fully modal attractor, to another time-periodic attractor, to a steady tetramodal attractor, to another time-periodic attractor having a full sequence of period-doublings (in the parameter space) to chaos. Numerous solutions are presented which provide conclusive evidence of the period-doubling cascades which precede chaos for this infinite-dimensional dynamical system. These results permit a computation of the lengths of subwindows which in turn provide an estimate for their successive ratios as the cascade develops. A calculation based on the numerical results is also presented to show that the period-doubling sequences found here for the Kuramoto-Sivashinsky equation, are in complete agreement with Feigenbaum's universal constant of 4.669201609.... Some preliminary work shows several other windows following the first chaotic one including periodic, chaotic, and a steady octamodal window; however, the windows shrink significantly in size to enable concrete quantitative conclusions to be made.This research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NASI-18605 while the authors were in residence at the Institute of Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support for the second author was provided by ONR Grant N-00014-86-K-0691 while he was at UCLA.     

Transition to chaos and escape phenomenon in two-degrees-of-freedom oscillator with a kinematic excitation

We study the dynamics of a two-degrees-of-freedom (two-DOF) nonlinear oscillator representing a quarter-car model excited by a road roughness profile. Modeling the road profile by means of a harmonic function, we derive the Melnikov criterion for a system transition to chaos or escape. The analytically obtained estimations are confirmed by numerical simulations. To analyze the transient vibrations, we used recurrences.     

On the escape of a resonantly excited couple of particles from a potential well

Nonlinear Dynamics - The escape dynamics of a damped system of two coupled particles in a truncated potential well under biharmonic excitation are investigated. It is assumed that excitation...