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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 958219, 27 pages
http://dx.doi.org/10.1155/2013/958219
Research Article

Multipulse Heteroclinic Orbits and Chaotic Dynamics of the Laminated Composite Piezoelectric Rectangular Plate

College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Received 21 March 2013; Accepted 21 May 2013

Academic Editor: Qingdu Li

Copyright © 2013 Minghui Yao and Wei Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates the multipulse global bifurcations and chaotic dynamics for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate by using an energy phase method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory, and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1 : 2 internal resonance and primary parametric resonance. The energy phase method is used for the first time to investigate the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The paper demonstrates how to employ the energy phase method to analyze the Shilnikov type multipulse heteroclinic bifurcations and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Numerical simulations show that for the nonlinear oscillations of the laminated composite piezoelectric rectangular plate, the Shilnikov type multipulse chaotic motions can occur. Overall, both theoretical and numerical studies suggest that chaos for the Smale horseshoe sense in motion exists.

1. Introduction

A piezoelectric material subjected to the mechanical force produces an electrical charge which is called the direct piezoelectric effect. Conversely, the material under an electrical field can generate mechanical stress which is called converse piezoelectric effect. In the development of intelligent structures, piezoelectric materials are widely designed as sensors and actuators for the active control of structures. Piezoelectric materials are usually clung to structural laminates used as devices to control deformation, shape, and vibration. There has been the model proposed for analysis of laminated composite plates containing active and passive piezoelectric layers. In aerospace applications, these smart structures are generally lightweight and have relatively large structural flexibility, and their flexibility can induce large deformation. Due to small material damping or the lack of environmental damping in space, fast motion or high-speed operation of laminated composite piezoelectric plates often generates the large amplitude vibration or geometrical nonlinearity. The complex nonlinear motions are strongly incited under the certain ranges of exciting frequency. Therefore, it is important to investigate large deformation and geometrically nonlinear effects of laminated composite piezoelectric plates in order to accurately design and effectively control structures.

Most of the studies on piezoelectric structures are based on linear piezoelectricity and linear elasticity theories, while research on the nonlinear dynamics of the piezoelectric structures is less. Lee et al. [1] incorporated the piezoelectric effect into the classical laminate plate theory and derived the distributed sensors and actuators which are capable of sensing and controlling the modal vibration of the cantilever plate. Tzou and Bao [2] formulated a new theory on thick anisotropic composite piezoelectric shell transducer laminates. This theory is applicable to a variety of composite piezoelectric structures. Reddy and Mitchell [3] developed geometrically nonlinear theories of laminated composite plates with piezoelectric laminae and used Hamilton's principle to derive equations of motion and boundary conditions of these theories. Hagood IV and McFarland [4] employed the Rayleigh-Ritz mode energy method to study the distributed piezoceramics and the traveling wave dynamics of the stator. Sadri et al. [5] presented the theoretical vibration model of the piezoelectric plate and used the Rayleigh-Ritz method to obtain the governing nonlinear equations of the piezoelectric plate. Vel and Batra [6] utilized Eshelby-Stroh formalism to analyze the quasistatic deformations of linear piezoelectric laminated plates. Yu and Hodges [7] took advantage of the variational-asymptotic method to construct a Reissner-Mindlin model for the laminated composite piezoelectric plates subjected to mechanical, thermal, and electric loads. Huang and Shen [8] exploited the higher order shear deformation plate theory and von Karman-type equation to investigate the nonlinear vibration and dynamic response of the laminated piezoelectric plates subjected to mechanical, electrical, and thermal loads. Della and Shu [9] gave a review on the various analytical models and numerical analyses for free vibration of delaminated composites including composite piezoelectric laminates under axial loads. Zhang et al. [10] made use of Reddy’s third-order shear deformation plate theory to analyze the bifurcations and chaotic dynamics of the four-edge simply supported, laminated composite piezoelectric rectangular plate in the case of 1 : 2 internal resonances. Sarangi and Ray [11] analyzed geometrically nonlinear transient vibrations of doubly curved laminated composite piezoelectric shells.

The global bifurcations and chaotic dynamics of high-dimensional nonlinear systems have been at the forefront of nonlinear dynamics for the past two decades. There are two ways of solutions on the Shilnikov type chaotic dynamics of high-dimensional nonlinear systems. One is Shilnikov type single-pulse chaotic dynamics and the other is the Shilnikov type multipulse chaotic dynamics. Most researchers focused on the Shilnikov type single-pulse chaotic dynamics of high-dimensional nonlinear systems. Wiggins [12] used the Melnikov method to investigate the global bifurcations and chaotic dynamics of three types high-dimensional perturbed Hamiltonian systems. With the aid of new global perturbation technique, Kovačič [13, 14] investigated the existence of the orbits homoclinic to resonance bands for the Hamiltonian systems and dissipative systems. Yagasaki [15] used the extended subharmonic Melnikov method and the modified homoclinic Melnikov method to examine periodic orbits and homoclinic motions in the coupled oscillators. Feng and Liew [16] canvassed the existence of the Shilnikov type single-pulse homoclinic orbits in the averaged equation which represents the modal interactions between two modes and influence of the fast mode on the slow mode. The global bifurcations and chaotic dynamics were investigated by Zhang et al. [17, 18] for the simply supported rectangular thin plates subjected to the parametrical-external excitation and the parametrical excitation. Yeo and Lee [19] employed the global perturbation technique to probe into the global dynamics of an imperfect circular plate with one-to-one internal resonance and obtained the criteria for chaotic motions of homoclinic orbits. Vakakis [20] adopted subharmonic and homoclinic Melnikov theory to study the strong nonlinear dynamic in a lattice and a lightweight of a nonlinear oscillator with nonlinearity stiffness.

Most research is on the Shilnikov type single-pulse global bifurcations and chaotic dynamics of high-dimensional nonlinear systems, but there are researchers investigating the Shilnikov type multipulse homoclinic and heteroclinic bifurcations and chaotic dynamics. So far, there are two theories of the Shilnikov type multipulse chaotic dynamics. One is the energy phase method and the other theory is the extended Melnikov method. Much achievement is made in the former theory of high-dimensional nonlinear systems. Haller and Wiggins [21] first established a simple energy-phase criterion which combined geometric singular perturbation theory, higher-dimensional Melnikov method, and transversality theory. Haller [22] derived a normal form for weak-strong resonance junctions in -degree-of-freedom, nearly integrable Hamiltonian systems. Haller [23] proposed a new unified theory of orbits homoclinic to resonance bands in a class of near-integrable dissipative systems. Subsequently, Haller and Wiggins [24, 25] further developed the energy phase method to examine the existence of the multipulse homoclinic orbits in the damped-forced nonlinear Schrodinger equation and perturbed the Hamiltonian systems. Haller and Wiggins [26] proved the existence of the Shilnikov type multipulse homoclinic orbits to invariant 3 spheres and utilized the energy-phase method to investigate chaotic dynamics near resonant equilibria in three-degree-of-freedom Hamiltonian systems. Haller [27] derived a universal homoclinic tree describing the bifurcations of the multipulse homoclinic orbits near the intersection of a weaker and a stronger resonance in -degree-of-freedom, nearly integrable Hamiltonian systems. Haller [28, 29] developed the energy-phase method to detect the existence of the Shilnikov-type multipulse orbits that repeatedly leave and come back to an invariant manifold with two different time scales in the perturbed nonlinear Schrodinger equation. Haller et al. [30] verified the existence of the Shilnikov-type multipulse homoclinic orbits to a spatially independent invariant torus in two coupled nonlinear Schrodinger equations with the damping and the quasiperiodical force. In book [31] published by Haller in 1999, they summarized the energy-phase method and presented a detailed procedure of the application to several problems in mechanics, which include the Shilnikov type multipulse homoclinic and heteroclinic orbits and chaotic dynamics.

Up to present, few researchers have made use of the energy phase method to study the Shilnikov type multipulse homoclinic and heteroclinic orbits and chaotic dynamics of high-dimensional nonlinear systems in engineering applications. Malhotra et al. [32] used the energy-phase method to investigate multipulse homoclinic orbits and chaotic dynamics for the motion of flexible spinning discs. McDonald and Namachchivaya [33] applied the Melnikov method, the Shilnikov method, and the energy-phase method to detect the presence of chaotic dynamics of parametrically excited pipes conveying fluid near a zero-to-one resonance. Yao and Zhang [34, 35] utilized the energy-phase method to analyze the Shilnikov type multipulse heteroclinic or homoclinic orbits and chaotic dynamics in a parametrically and externally excited rectangular thin plate and a laminated composite piezoelectric rectangular plate. Yu and Chen [36, 37] made use of the energy-phase method to examine the Shilnikov type multipulse homoclinic orbits of a nonlinear cyclic system and a harmonically excited circular plate. Zhang et al. [38] extended the energy-phase method to six-dimensional system from four-dimensional system. They adopted the energy-phase method for six-dimensional system to delve into multipulse chaotic dynamics of a composite laminated piezoelectric rectangular plate subjected to the transverse, in-plane excitations and the piezoelectric excitation.

The study on the second theory of the Shilnikov type multipulse chaotic dynamics was stated by Kovačič and Wettergren [39]. They presented the extended Melnikov method to investigate the existence of the multipulse jumping of homoclinic orbits and chaotic dynamics in resonantly forced coupled pendula. Furthermore, Kaper and Kovačič [40] studied the existence of several classes of the multibump orbits homoclinic to resonance bands for completely integrable Hamiltonian systems subjected to small Hamiltonian amplitude and damped perturbations. Camassa et al. [41] applied the extended Melnikov method to analyze the multipulse jumping of homoclinic and heteroclinic orbits in a class of perturbed Hamiltonian systems. Until recently, Zhang and Yao [42, 43] introduced the extended Melnikov method to the engineering field. They came up with a simplification of the extended Melnikov method in the resonant case and utilized it to analyze the Shilnikov type multipulse 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. Yao et al. [44] made use of the extended Melnikov method and numerical method to investigate multipulse chaotic dynamics in nonplanar motion of parametrically excited viscoelastic moving belt.

Despite extensive research of nonlinear dynamics in the laminated composite piezoelectric plate, multipulse chaotic dynamics has been studied rarely. Few researchers utilized the energy-phase method to investigate the Shilnikov type multipulse chaotic dynamics of high-dimensional nonlinear systems in engineering applications in the past several years. We have previously studied homoclinic bifurcations and multipulse chaotic dynamics of the laminated composite piezoelectric plate under the case of 1 : 3 internal resonances by applying the energy-phase method. In this paper, we have used the energy-phase method to investigate heteroclinic bifurcations and multipulse chaotic dynamics of the laminated composite piezoelectric plate under the case of 1 : 2 internal resonances.

It is the purpose of this paper to fill the research gap by investigating the Shilnikov type multipulse chaotic dynamics in the complex motion of the laminated composite piezoelectric plate. Based on the von Karman type equations and the Reddy’s third-order shear deformation plate theory, the Hamilton’s principle is employed to obtain the governing nonlinear equations of the laminated composite piezoelectric rectangular plate with combined parametric excitation and transverse load. We apply the method of multiple scales and Galerkin’s approach to the partial differential governing equations to obtain the four-dimensional averaged equation for the case of 1 : 2 internal resonances and primary parametric resonance. From the averaged equation, the theory of normal form is used to find the explicit formulas of normal form. Finally, we employ the energy-phase method to analyze the Shilnikov type multipulse orbits and chaotic dynamics in the laminated composite piezoelectric plate. The analysis indicates that there exist the Shilnikov type multipulse jumping orbits in the perturbed phase space for the averaged equations. The results from numerical simulation also show that the chaotic motion can occur in the motion of the laminated composite piezoelectric plate, which verifies the analytical prediction. The Shilnikov type multipulse orbits are discovered from the results of numerical simulation. In summary, both theoretical and numerical studies demonstrate that chaos for the Smale horseshoe sense in the motion exists.

2. Equations of Motion and Perturbation Analysis

We consider a four-edge simply supported laminated composite piezoelectric rectangular plate, where the length, the width, and the thickness are denoted by a, b, and h, respectively. The laminated composite piezoelectric rectangular plate is subjected to in-plane excitation, transverse excitation, and piezoelectric excitation, as shown in Figure 1. We consider the laminated composite piezoelectric rectangular plate as regular symmetric cross-ply laminates with layers with respect to principal material coordinates alternatively oriented at and to the laminate coordinate axes. Some of the layers are made of the piezoelectric materials as actuators, and the other layers are made of fiber-reinforced composite materials. It is assumed that different layers of the symmetric cross-ply composite laminated piezoelectric rectangular plate are perfectly clung to each other, and piezoelectric actuator layers are embedded in the plate. The fiber direction of odd-numbered layers is the -direction of the laminate. The fiber direction of even-numbered layers is the -direction of the laminate. Simply supported plate with immovable edges satisfies the symmetry requirement that eliminates the coupling between bending and extension. However, the displacement of is free to move at the edge of , and the displacement of is free to move at the edge of . Therefore, the membrane stress is smaller and there exists the coupling between bending and extension. A Cartesian coordinate system is located on the middle surface of the composite laminated piezoelectric rectangular plate. Assume that and describe the displacements of an arbitrary point and a point on the middle surface of the composite laminated piezoelectric rectangular plate in the , , and directions, respectively. It is also assumed that in-plane excitations of the composite laminated piezoelectric rectangular plate are loaded along the -direction at and the -direction at with the form of and , respectively. Transverse excitation loaded to the composite laminated piezoelectric rectangular plate is expressed as . The dynamic electrical loading is represented by .

958219.fig.001
Figure 1: The model of a laminated composite piezoelectric rectangular plate is given.

In this paper, Reddy’s third-order shear deformation description of the displacement field is adopted:where () are the deflection of a point on the middle surface, are the displacement components along the () coordinate directions, and and denote the rotation components of normal to the middle surface about the and axes, respectively.

The nonlinear strain-displacement relations are assumed to have the following form:

Stress constitutive relations are presented as follows: where and denote the mechanical stresses and strains in extended vector notation, represents the elastic stiffness tensor, stands for the electric field vector, and is the piezoelectric tensor.

According to Hamilton’s principle, the nonlinear governing equations of motion for the composite laminated piezoelectric rectangular plate are given in the previou studies as follows [10, 35]:where the dot represents the partial differentiation with respect to time , the comma denotes the partial differentiation with respect to a specified coordinate, is the damping coefficient, and all kinds of inertias in (4a), (4b), (4c), (4d), and (4e) are calculated by

In addition, the stress resultants are represented as follows:where () represent the piezoelectric stress resultants, and are piezoelectric constants, denotes electric field, and , , , , , and are, respectively, the stiffness elements of the laminated composite piezoelectric rectangular plate, which are denoted as

Substituting (6a), (6b), (6c), (6d), (6e), (6f), (6g), (6h), (6i), (6j), (6k), (6l), and (6m) into (4a), (4b), (4c), (4d), and (4e) yields governing equations of motion in terms of generalized displacements as follows:

The simply supported boundary conditions of the composite laminated piezoelectric rectangular plate can be represented as follows [10, 35, 45]:

The boundary condition (9f) includes the influence of the in-plane load. We consider complicated nonlinear dynamics of the composite laminated piezoelectric rectangular plate in the first two modes of , , , , and . It is desirable that we select an appropriate mode function to satisfy the boundary condition. Thus, we can write , , , , and in the following forms:

Based on the study given by Nosir and Reddy and Bhimaraddi [46, 47], we neglect the in-plane and rotary inertia terms in (8a), (8b), (8d), and (8e) since their influences are small compared to the transverse inertia term. By means of the Galerkin method, substituting (10a), (10b), (10c), (10d), and (10e) into (8a), (8b), (8c), (8d), and (8e), integrating and neglecting all inertia terms in (8a), (8b), (8d), and (8e), we obtain the expressions of , , , , , , , and in terms of and as follows:where the coefficients presented in (11a), (11b), (11c), (11d), (11e), and (11f) can be found in Appendix A.

In order to obtain the dimensionless governing equations of motion, we introduce the transformations of the variables and parameters:

Based on the practical work condition of laminated composite rectangular plates, and theoretical and experimental studies obtained by Reddy and Mitchell [3, 45], it is known that the nonlinear transverse vibration of laminated composite rectangular plates occupies the main aspect of the dynamical characteristics. The transverse vibration is far greater than the other additional vibrations in the and directions, respectively. Therefore, the equations for and can be neglected. We mainly consider the nonlinear transverse vibration of laminated composite rectangular plates.

For simplicity, we drop the overbar in the following analysis. Substituting ((9a), (9b), (9c), (9d), (9e), and (9f))–(12) into (8c) and applying the Galerkin procedure, we obtain the governing equations of motion of the composite laminated piezoelectric rectangular plate for the dimensionless as follows:where the coefficients presented in (13a) and (13b) are given in Appendix B.

The above equations include the cubic terms, in-plane excitation, transverse excitation, and piezoelectric excitation. Equations (13a) and (13b) can describe the nonlinear transverse oscillations of the composite laminated piezoelectric rectangular plate. We only study the case of primary parametric resonance and 1 : 2 internal resonances. In this resonant case, there are the following resonant relations: where and are two detuning parameters.

The method of multiple scales [48] is employed in (13a) and (13b) to find the uniform solutions in the following form:where , .

Substituting (14), (15a), and (15b) into (13a) and (13b) and balancing the coefficients of the corresponding powers of on the left-hand and right-hand sides of equations, the four-dimensional averaged equations in the Cartesian form are obtained as follows:

3. Computation of Normal Form

In order to assist the analysis of the Shilnikov type multipulse orbits and chaotic dynamics of the laminated composite piezoelectric rectangular plate, it is necessary to reduce the averaged equations (16a), (16b), (16c), and (16d) to a simpler normal form. It is found that there are and symmetries in the averaged equations (16a), (16b), (16c), and (16d) without the parameters. Therefore, these symmetries are also held in normal form.

We take into account the excitation amplitude as a perturbation parameter. Amplitude can be considered as an unfolding parameter when the Shilnikov type multipulse orbits are investigated. Obviously, when we do not consider the perturbation parameter, (16a), (16b), (16c), and (16d) becomewhere .

It is obviously known that (17a), (17b), (17c), and (17d) have a trivial zero solution at which the Jacobian matrix can be written as

The characteristic equation corresponding to the trivial zero solution is

Let When , , and are satisfied simultaneously, (17a), (17b), (17c), and (17d) have one nonsemisimple double zero and a pair of pure imaginary eigenvalues: where .

Considering , , , and as the perturbation parameters, letting , and setting , then, averaged equations (17a), (17b), (17c), and (17d) without the perturbation parameters become the following forms:

According to (22a), (22b), (22c), and (22d), we have

Executing the Maple program given by Zhang et al. [49], the 3-order normal form of (22a), (22b), (22c), and (22d) is obtained as

The nonlinear transformation used here is given as follows:

The above nonlinear transformation is computed through the Maple program given by Zhang et al. [49], and completely agrees with those presented by Yu et al. [50]. Therefore, a simpler 3-order normal form with the parameters for the averaged equations (16a), (16b), (16c), and (16d) is obtained as follows:where the coefficients are , , , and , respectively.

Further, let Substituting (27) into (26a), (26b), (26c), and (26d) yields

In order to get the unfolding of (28a), (28b), (28c), and (28d), a linear transformation is introduced:

Substituting (29) into (28a), (28b), (28c), and (28d), and canceling nonlinear terms including the parameter yield the unfolding as follows:where , , , and .

The scale transformations to be introduced into (30a), (30b), (30c), and (30d) are Then, the normal forms (30a), (30b), (30c), and (30d) can be rewritten in the form with the perturbations as follows:where the Hamiltonian function is of the following form: and , , , and are the linear perturbation terms induced by the dissipative effects:

4. Heteroclinic Bifurcations

In this section, we focus on studying the nonlinear dynamics characteristic of the unperturbed system. When , it can be seen that the system from (32a), (32b), (32c), and (32d) is an uncoupled two-degree-of-freedom nonlinear system. The variable in the subspace of (32a), (32b), (32c), and (32d) becomes a parameter since . Consider the first two decoupled equations of (32a), (32b), (32c), and (32d),

Since , (35a) and (35b) can exhibit the heteroclinic bifurcations. It is obvious from (35a) and (35b) that when , the only solution to (35a) and (35b) is the trivial zero solution, , which is the saddle point. On the curve defined by , that is, or the trivial zero solution bifurcates into three solutions through a pitchfork bifurcation, which are given by and , respectively, where

From the Jacobian matrix evaluated at the nonzero solutions, it can be found that the singular points are the saddle points. It is observed that and actually represent the amplitude and the phase of vibrations, respectively. Therefore, we assume that and (37) become such that for all , (35a) and (35b) have two hyperbolic saddle points, , which are connected by a pair of heteroclinic orbits, , that is, . Thus, in the full four-dimensional phase space, the set defined by is a two-dimensional invariant manifold. From the results obtained by Kovačič [13, 14], it is known that the two-dimensional invariant manifold is normally hyperbolic. The two-dimensional normally hyperbolic invariant manifold has the three-dimensional stable and unstable manifolds represented as and , respectively. The existence of the heteroclinic orbit of (35a) and (35b) to indicates that and intersect nontransversally along a three-dimensional heteroclinic manifold denoted by , which can be written as

We analyze the dynamics of the unperturbed system of (32a), (32b), (32c), and (32d) restricted to . Considering the unperturbed system of (32a), (32b), (32c), and (32d) restricted to yieldswhere

From the results obtained by Kovačič [13, 14], it is known that if ; then constant is called a periodic orbit, and if ; then constant is known as a circle of the singular points. Any value of at which is a resonant value and these singular points are resonant singular points. We denoted a resonant value by such that Then, we obtain

Figure 2 shows the geometry structure of the stable and unstable manifolds of in the full four-dimensional phase space for the unperturbed system of (32a), (32b), (32c), and (32d). Since represents the phase of oscillations, when , the phase shift of oscillations is defined by

fig2
Figure 2: The geometric structure of manifolds ; and are given in the full four-dimensional phase space.

The physical interpretation of the phase shift is the phase difference between the two end points of the orbit. In the subspace , there exists a pair of the heteroclinic orbits connecting to the two saddles. Therefore, the homoclinic orbit in subspace represents, in fact, a heteroclinic connecting in the full four-dimensional space . The phase shift denotes the difference of the value as a trajectory leaving and returning to the basin of the attraction of . We will use the phase shift in the subsequent analysis to obtain the condition for the existence of the Shilnikov type multipulse orbit. The phase shift will be calculated later in the heteroclinic orbit analysis.

We consider the heteroclinic orbits of (35a) and (35b). Let and let ; then (35a) and (35b) can be rewritten asSet ; then (47a) and (47b) are a system with the Hamiltonian function:

When , there exists a heteroclinic loop which consists of the two hyperbolic saddles and a pair of heteroclinic orbits . In order to calculate the phase shift and the energy difference function, we obtain the equations of a pair of heteroclinic orbits given by

We turn our attention to the computation of the phase shift. Substituting the first equation of (49a) and (49b) into the fourth equation of the unperturbed system of (32a), (32b), (32c), and (32d) yields Integrating (50) yields where .

At , there is . Therefore, the phase shift may be expressed as

5. Dissipative Perturbations of Homoclinic Loop

In this section, the effects of small perturbation terms on the unperturbed system are analyzed in detail. We now analyze dynamics of the perturbed system and the effect of small perturbations on . Based on the analysis given by Kovačič [13, 14], it is known that along with its stable and unstable manifolds is invariant under small, sufficiently differentiable perturbations. It is noticed that in (35a) and (35b) maintain the characteristic of the hyperbolic singular point under small perturbations, in particular, . Therefore, we obtain

Considering the last two equations of (32a), (32b), (32c), and (32d) yieldsIt is known from the above analysis that the last two equations of (32a), (32b), (32c), and (32d) are of a pair of pure imaginary eigenvalues. Therefore, the resonance can occur in (54a) and (54b). Also introduce the scale transformations

Substituting the above transformations into (54a) and (54b) yieldswhere .

When , (56a) and (56b) become

The unperturbed system from (57a) and (57b) is a Hamilton system with the following function: The singular points of (57a) and (57b) are represented as

Based on the characteristic equations evaluated at the two singular points and , we can know the stabilities of these singular points. The Jacobian matrix of (57a) and (57b) is The characteristic equation corresponding to the singular point is obtained as

When the condition is satisfied, (57a) and (57b) have a pair of pure imaginary eigenvalues. Therefore, it is known that the singular point is a center point.

The characteristic equation corresponding to the singular point is represented by When the condition is satisfied, (57a) and (57b) have two real, unequal, and opposite sign eigenvalues. Therefore, the singular point is a saddle which is connected to itself by a homoclinic orbit. The phase portrait of the system from (57a) and (57b) is shown in Figure 3(a).

fig3
Figure 3: Dynamics on the normally hyperbolic manifold is described; (a) the unperturbed case; (b) the perturbed case.

It is found that for the sufficiently small parameter , the singular point remains a hyperbolic singular point of the saddle stability type. It is known that the Jacobian matrix of the linearization of (56a) and (56b) is of the following form: or

Based on (64), we find that the leading order term of the trace in the linearization of (56a) and (56b) is less than zero inside the homoclinic loop. Therefore, for small perturbations, the singular point becomes a hyperbolic sink . The phase portrait of the perturbed system from (56a) and (56b) is also depicted in Figure 3(b).

Using the function (58), at , the estimate of the basin of attractor for is obtained as Substituting in (59) into (65) yields

Define an annulus near as where is a constant and is sufficiently large so that the unperturbed homoclinic orbit is enclosed within the annulus.

It is noticed that the three-dimensional stable and unstable manifolds of , denoted as and , are subsets of and , respectively. We will indicate that for the perturbed system, the saddle focus on has the multipulse orbits which come out of the annulus and can return to the annulus in the full four-dimensional space. These orbits, which are asymptotic to some invariant manifolds in the slow manifold , leave and enter a small neighborhood of multiple times, and, finally, return and approach an invariant set in asymptotically, as shown in Figure 4. In Figure 4, this is an example of the three-pulse jumping orbit which depicts the formation mechanism of the multipulse orbits.

958219.fig.004
Figure 4: The Shilnikov type three-pulse homoclinic orbits are obtained.

6. Energy Difference Function

In this section, the energy-phase method developed by Haller and Wiggins [2131] is utilized to discover the existence of the Shilnikov type multipulse orbits and chaotic dynamics of nonlinear vibration for the laminated composite piezoelectric rectangular plate with combined parametric excitation and transverse load. The energy-phase method, which was first presented by Haller and Wiggins [21], is a new global perturbation method, which is different from the global perturbation technique developed by Kovačič and Wiggins [1214]. Haller and Wiggins [2131] gave the unperturbed system expressed by a mixture of action-angle variables, which can describe homoclinic and heteroclinic behavior. The key feature of their analysis is that the unperturbed system has a resonance with the action-angle variables. The unperturbed integrable system contains a normally hyperbolic invariant two-dimensional manifold, which has three-dimensional stable and unstable manifolds coinciding along a branch. Action-angle variables illustrate the dynamics on the normally hyperbolic two-dimensional manifold. The situation of the resonance leads to a circle of fixed points on the two-dimensional manifold. Two different points on the circle of fixed points are connected by a heteroclinic orbit. These heteroclinic orbits consist of part of a foliation of the three-dimensional stable and unstable manifolds on the normally hyperbolic two-dimensional manifold. Obviously, the perturbation can dramatically alter the dynamics near the circle of fixed points. Thus, the dynamics restricted to the normally hyperbolic two-dimensional manifold near the resonance becomes hyperbolic fixed points with periodic orbits surrounding homoclinic or heteroclinic trajectories. Each homoclinic or heteroclinic trajectory connects with the hyperbolic points. Haller and Wiggins [2131] gave conditions for the existence of homoclinic or heteroclinic orbits created near the resonance in the full four-dimensional phase space.

The energy-phase method has three steps in the process in this paper. The first step involves analyzing the perturbed dynamics restricted to the normally hyperbolic invariant two-dimensional manifold near the resonance. The used approach is based on nonlinear oscillation theory. The second step requires manifesting the existence of heteroclinic orbits to the normally hyperbolic invariant two-dimensional manifold. A higher-dimensional Melnikov theory is mainly used in the analysis process of this step. The final step is the most difficult and represents the most innovative part of the energy-phase method. Based on the Haller and Wiggins study in [2131], we prove the existence of multipulse heteroclinic orbits to specific orbits on the two-dimensional manifold in this step. The analysis is complicated by perturbations of nontransversal intersections of manifolds. According to the investigation given by Haller and Wiggins [2131], we use the geometric singular perturbation theory of Fenichel to analyze the existence of heteroclinic orbits in the full four-dimensional phase space. Based on foliations of stable and unstable manifolds, Fenichel’s theory combines with energy-type arguments which are suited for the Hamiltonian systems. The analysis shows that the existence of heteroclinic orbits depends only on an energy-phase criterion which is obtained from a reduced, one-degree-of-freedom Hamiltonian system. Therefore, the energy-phase method is combined with geometric singular perturbation theory, higher-dimensional Melnikov method, and transversality theory.

The energy-phase method can be utilized to detect the Shilnikov type multipulse heteroclinic orbits to slow manifolds of near-integrable four-dimensional or higher-dimensional nonlinear systems. The key to energy-phase method is the calculation of the energy difference function, which is actually the difference of the Hamiltonian function. Based on the higher-dimensional Melnikov theory and the transversality theory, Haller and Wiggins [21] derived the expression of the energy difference function. In order to illustrate the existence of the multipulse orbits, it is important to obtain the expression of the energy difference function. Using the general expression derived by Haller and Wiggins [2131] for the class of systems, the energy difference function for the dissipative case is given as follows: where