Multi-pulse Homoclinic Orbits Research Articles

Stability and chaotic dynamic analysis of Li-doped fullerene-IRMOF composite materials for hydrogen storage

Isoreticular metal organic framework (IRMOF) material is a type of ideal hydrogen storage material for its porosity and large surface area. Lithium doping and fullerene impregnation can enhance the hydrogen storage capacity of IRMOF efficiently. In this paper, the nonlinear dynamic characteristics of Li-doped fullerene-IRMOF composite material are qualitatively discussed in theory, which determine its thermal stability. A type of Li-doped fullerene-IRMOF composite material is proposed for hydrogen storage, and a new constitutive model is proposed to describe the hysteretic property of IRMOF. The system's nonlinear dynamic model is developed, and the fast-slow dynamic analysis method is introduced to interpret not only the system's jumping phenomena between the multi-pulse homoclinic and heteroclinic orbits but also its chaotic dynamics. The nonlinear dynamic responses of the system under deterministic and stochastic excitation are obtained respectively, and symmetrical and asymmetric orbital fusion phenomena have been discovered. The results of theoretical analysis and numerical simulation both show that the system has novel nonlinear dynamic characteristics such as double periodic motion, multi-pulse jumping orbits, and chaos; multi-pulse jumping phenomena are caused by the stiffness difference between fullerene and IRMOF, which form a fast-slow system; symmetrical fusion and asymmetrical fusion of system's orbits occur under different parameters. Safe, efficient and rapid hydrogen storage and release can be achieved through controlling the system's parameters to adjust the dynamic behavior of the system, which will be a development direction in the application of hydrogen energy.

Chaotic Dynamics of Non-Autonomous Nonlinear System for a Sandwich Plate with Truss Core

This paper deals with the multi-pulse chaotic dynamics of a sandwich plate with truss core under transverse and in-plane excitations. In order to analyze the complicated nonlinear behaviors of the sandwich plate model by means of the improved extended Melnikov technique, the two-degrees non-autonomous system is transformed into an appropriate form. Through theoretical analysis, the sufficient conditions for the existence of multi-pulse homoclinic orbits and the criterion for the occurrence of chaotic motion are obtained. The damping coefficients and transverse excitation parameters are considered as the control parameters to discuss chaotic behaviors of the sandwich plate system. Numerical results and the maximal Lyapunov exponents are performed to further test the existence of the multi-pulse jumping orbits. The theoretical predictions and numerical results demonstrate that the chaos phenomena may exist in the parametrical excited sandwich plate.

Chaotic Dynamics of Piezoelectric Composite Laminated Beams

Chaos in piezoelectric composite laminated beams has significant implications in the design of this model. Some results for this model have been obtained numerically. With the energy-phase method and numerical simulations, global dynamics of piezoelectric composite laminated beams is investigated in this paper. The average equation of the piezoelectric composite laminated beam is obtained by the normal form theory. The existence of multipulse homoclinic orbits for undisturbed and dissipative cases is analyzed by the energy-phase method, and the mechanism of chaotic motion of the system is given. The effect of the dissipation factor on pulse sequence and layer radius is studied in detail. The chaotic motion of the system is verified by numerical simulations.

Multipulse Homoclinic Orbits and Chaotic Dynamics of a Reinforced Composite Plate with Carbon Nanotubes

The multipulse homoclinic orbits and chaotic dynamics of a reinforced composite plate with the carbon nanotubes (CNTs) under combined in-plane and transverse excitations are studied in the case of 1 : 1 internal resonance. The method of multiple scales is adopted to derive the averaged equations. From the averaged equations, the normal form theory is applied to reduce the equations to a simpler normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method proposed by Haller and Wiggins is utilized to examine the global bifurcations and chaotic dynamics of the CNT-reinforced composite plate. The analytical results demonstrate that the multipulse Shilnikov-type homoclinic orbits and chaotic motions exist in the system. Homoclinic trees are constructed to illustrate the repeated bifurcations of multipulse solutions. In order to verify the theoretical results, numerical simulations are given to show the multipulse Shilnikov-type chaotic motions in the CNT-reinforced composite plate. The results obtained here imply that the motion is chaotic in the sense of the Smale horseshoes for the CNT-reinforced composite plate.

Global dynamics of a flexible asymmetrical rotor

Global dynamics of a flexible asymmetrical rotor resting on vibrating supports is investigated. Hamilton’s principle is used to derive the partial differential governing equations of the rotor system. The equations are then transformed into a discretized nonlinear gyroscopic system via Galerkin’s method. The canonical transformation and normal form theory are applied to reduce the system to the near-integrable Hamiltonian standard forms considering zero-to-one internal resonance. The energy-phase method is employed to study the chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of damping dissipative perturbation, the existence of generalized Silnikov-type multi-pulse orbits which are homoclinic to fixed points on the slow manifold is examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes is obtained analytically. The global results are finally interpreted in terms of the physical motion of axis orbit. The present study indicates that the existence of multi-pulse homoclinic orbits provides a mechanism for how energy may flow from the high-frequency mode to the low-frequency mode when the rotor system operates near the first-order critical speed.

Multi-Pulse Chaotic Dynamics of Circular Mesh Antenna with 1:2 Internal Resonance

The multi-pulse homoclinic orbits and chaotic dynamics of an equivalent circular cylindrical shell for the circular mesh antenna are investigated in the case of 1:2 internal resonance in this paper for the first time. Applying the method of averaging, the four-dimensional averaged equation in the Cartesian form is obtained. The theory of normal form is used to reduce the averaged equation to a simpler form. Based on the simplified system, the energy phase method is employed to investigate the homoclinic bifurcations and the Shilnikov type multi-pulse chaotic dynamics. First, the energy difference function and the zeroes of the energy difference function are obtained. Then, the existence of the Shilnikov type multi-pulse orbits is determined. The homoclinic trees are depicted to describe the relationship among the layers diameter, the pulse numbers and the phase shift. Finally, we need to verify the condition which makes sure that any multi-pulse orbit departing from a slow sink comes back to the domain of attraction of one of the sinks. The results obtained here show the existence of the Shilnikov type multi-pulse chaotic motions of the circular mesh antenna. Numerical simulations are used to find multi-pulse chaotic motions. The results of the theoretical analysis are in qualitative agreement with the results obtained using numerical simulation.

Global dynamics of composite panels with free-layer damping treatment in subsonic flow

Global dynamics of forcedly excited composite panels with free layer damping treatment in subsonic flow near the first-order critical velocity is investigated. Hamilton’s principle is implemented to derive the PDE of such fluid-structure interaction systems. Then the governing equation is transformed into a discretized nonlinear gyroscopic system via assumed modes and Galerkin’s method. The canonical transformations and normal form theory are applied to reduce the equations of motion to near-integrable Hamiltonian standard forms considering zero to one internal resonance. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. In both the Hamiltonian and the dissipative perturbation case, the homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are demonstrated. In the case of dissipative perturbation, the existence of generalized Šilnikov’s type of orbits which are homoclinic to fixed points on the slow manifold are examined and the parameter region for which the dynamical system may exhibit chaotic motions in the sense of Smale horseshoes are obtained analytically. The present research illustrates that the existence of multi-pulse homoclinic orbits can provide a mechanism for how energy flow from the high-frequency mode to the low-frequency mode. The global results are finally interpreted in terms of the physical traveling wave motion of such gyroscopic continua.

Homoclinic orbits and chaos of a supercritical composite panel with free-layer damping treatment in subsonic flow

Multi-pulse homoclinic orbits and chaotic dynamics of a supercritical composite panel with free layer damping treatment in subsonic flow are investigated considering one to two internal resonance. Inviscid potential flow theory is employed to exhibit the aerodynamic pressure and Kelvin’s model is used to describe the viscoelastic property of the free damping layer. By Hamilton’s principle, the governing equation of the composite panel in the subcritical regime is derived. In the supercritical regime, the buckling configuration is solved analytically and the PDE is obtained by introducing a displacement transformation for nontrivial equilibrium configuration. Then the governing equation in the first supercritical region is transformed into a discretized nonlinear gyroscopic system via assumed modes and then Galerkin’s method. The method of multiple scales and canonical transformation are applied to reduce the equations of motions to the near-integrable Hamiltonian standard forms. The Energy-Phase method is employed to demonstrate the existence of chaotic dynamics by identifying the existence of multi-pulse jumping orbits in the perturbed phase space. The global solutions are finally interpreted in terms of the physical motion of the gyroscopic continua and the dynamical mechanism of chaotic pattern conversion between the forward traveling wave motion and the complex bidirectional traveling wave motion are discussed.

Multi-pulse orbits and chaotic dynamics of a symmetric cross-ply composite laminated cantilever rectangular plate

The multi-pulse homoclinic orbits and chaotic dynamics of a symmetric cross-ply composite laminated cantilever rectangular plate under in-plane and moment excitations are investigated with 1:2 internal resonance. Based on the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues, the energy-phase method proposed by Haller and Wiggins is employed to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the composite laminated cantilever rectangular plate. The analysis of the global dynamics indicates that there exist the Shilnikov-type multi-pulse jumping orbits homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. Homoclinic trees that describe the repeated bifurcations of multi-pulse solutions are presented. It can be found a gradual breakup of the homoclinic tree and the reducing of pulse numbers for the jumping homoclinic orbits if the dissipation factor is increased. The chaotic motions of the symmetric cross-ply composite laminated cantilever plate are also found by using the numerical simulation.

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper, the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied. The damping, parametrical excitation and the nonlinearities are regarded as weak. The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales. The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method. We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos. At last, numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

AbstractWe consider the existence of spots and stripes for a class of nonlinear Schrödinger‐type equations in the presence of nearly one‐dimensional localized potentials. Under suitable assumptions on the potential, we construct various types of waves that are localized in the direction of the potential and have single‐ or multihump, or periodic profile in the perpendicular direction. The analysis relies upon a spatial dynamics formulation of the existence problem, together with a center manifold reduction. This reduction procedure allows these waves to be realized as unipulse or multipulse homoclinic orbits, or periodic orbits in a reduced system of ordinary differential equations. Copyright © 2013 John Wiley & Sons, Ltd.

Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam

The multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam are investigated in the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method is employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, leading to chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures.

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.

GLOBAL BIFURCATIONS AND CHAOTIC DYNAMICS IN SUSPENDED CABLES

The global bifurcations and chaotic dynamics of parametrically and externally excited suspended cables are investigated in this paper. The governing equations are obtained to describe the nonlinear transverse vibrations of suspended cables. The Galerkin procedure is introduced to simplify the governing equations of motion to ordinary differential equations with two-degrees-of-freedom. The case of one-to-one internal resonance between the modes of suspended cables, primary and principal parametric resonance of suspended cables is considered. With the method of multiple scales, parametrically and externally excited system is transformed to the averaged equation, based on which, the recently developed global bifurcation method is employed to detect the presence of orbits which are homoclinic to certain invariant sets for the resonant case. The analysis of the global bifurcations indicates that there exist the generalized Šhilnikov type multipulse homoclinic orbits in the averaged equation of suspended cables. The results obtained here mean that chaotic motions can occur in suspended cables. Numerical simulations also verify the analytical predictions. It is found, according to the results of numerical simulations, that the Šhilnikov type multipulse homoclinic orbits exist in the nonlinear motion of the cables.

On the Maslov index of multi-pulse homoclinic orbits

Multi-pulse homoclinic orbits of Hamiltonian systems on can be classified by a sequence of integers. In this paper, we find the surprising result that this string of integers encodes the value of the Maslov index of the homoclinic orbit. Our results include a computable formulation of the Maslov index for homoclinic orbits and a robust numerical method for the evaluation of the Maslov index.

THEORIES OF MULTI-PULSE GLOBAL BIFURCATIONS FOR HIGH-DIMENSIONAL SYSTEMS AND APPLICATION TO CANTILEVER BEAM

The aim of this survey paper is to illustrate the perspectives on the theories of the single- and multi-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems and applications to several engineering problems in the past two decades. Two main methods for studying the Shilnikov type multi-pulse homoclinic and heteroclinic orbits in high-dimensional nonlinear systems, which are the energy-phase method and generalized Melnikov method, are briefly demonstrated in the theoretical frame. In addition, the theory of normal form and an improved adjoint operator method for high-dimensional nonlinear systems is also applied to describe a reducing procedure to high-dimensional nonlinear systems. The aforementioned methods are utilized to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of the cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end. How to employ these methods to analyze the Shilnikov type multi-pulse homoclinic and heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example.

Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam

The aim of this paper is to investigate the multi-pulse global bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of a cantilever beam subjected to a harmonic axial excitation and two transverse excitations at the free end by using an extended Melnikov method in the resonant case. First, the extended Melnikov method for studying the Shilnikov-type multi-pulse homoclinic orbits and chaos in high-dimensional nonlinear systems is briefly introduced in the theoretical frame. Then, this method is utilized to investigate the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics for the nonlinear non-planar oscillations of the cantilever beam. How to employ this method to analyze the Shilnikov-type multi-pulse homoclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications is demonstrated through this example. Finally, the results of numerical simulation are given and also show that the Shilnikov-type multi-pulse chaotic motions can occur for the nonlinear non-planar oscillations of the cantilever beam, which verifies the analytical prediction.

Homoclinic bifurcations at the onset of pulse self-replication

We establish a series of properties of symmetric, N-pulse, homoclinic solutions of the reduced Gray–Scott system: u ″ = u v 2 , v ″ = v − u v 2 , which play a pivotal role in questions concerning the existence and self-replication of pulse solutions of the full Gray–Scott model. Specifically, we establish the existence, and study properties, of solution branches in the ( α , β ) -plane that represent multi-pulse homoclinic orbits, where α and β are the central values of u ( x ) and v ( x ) , respectively. We prove bounds for these solution branches, study their behavior as α → ∞ , and establish a series of geometric properties of these branches which are valid throughout the ( α , β ) -plane. We also establish qualitative properties of multi-pulse solutions and study how they bifurcate, i.e., how they change along the solution branches.

Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt

Abstract In this paper, the Shilnikov type multi-pulse orbits and chaotic dynamics of parametrically excited viscoelastic moving belt are studied in detail. Using Kelvin-type viscoelastic constitutive law, the equations of motion for viscoelastic moving belt with the external damping and parametric excitation are given. The four-dimensional averaged equation under the case of primary parametric resonance is obtained by directly using the method of multiple scales and Galerkin’s approach to the partial differential governing equation of viscoelastic moving belt. From the averaged equations obtained here, the theory of normal form is used to give the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on normal form, the energy-phrase method is employed to analyze the global bifurcations and chaotic dynamics in parametrically excited viscoelastic moving belt. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type multi-pulse homoclinic orbits in the averaged equation. The results obtained above mean the existence of the chaos for the Smale horseshoe sense in parametrically excited viscoelastic moving belt. The chaotic motions of viscoelastic moving belts are also found by using numerical simulation. A new phenomenon on the multi-pulse jumping orbits is observed from three-dimensional phase space.

Multi-Pulse Homoclinic Orbits and Chaotic Dynamics in Motion of Parametrically Excited Viscoelastic Moving Belt

Article Multi-Pulse Homoclinic Orbits and Chaotic Dynamics in Motion of Parametrically Excited Viscoelastic Moving Belt was published on March 1, 2005 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 6, issue 1).