Multi-pulse Orbits and Homoclinic Trees in a Non-autonomous Resonant Hamiltonian System

In this study, we develop the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems. Our generalized energy-phase method applies to integrable, two-degree-of freedom non-autonomous resonant Hamiltonian systems. As an example, we investigate the multi-pulse orbits and homoclinic trees for a parametrically excited, simply supported rectangular thin plate of two-mode approximation. In both the Hamiltonian and dissipative case we fin d homoclinic trees, which describe the repeated bifurcations of multi -pulse solutions, and we present visualizations of these complicated structures.

Although the energy-phase method has been applied widely to engineering problems, it was used to solve autonomous perturbed Hamiltonian systems.It is worth to mention that the energy-phase method has never used in non-autonomous nonlinear dynamical systems.In this paper, we develop the energy-phase method to deal with non-autonomous nonlinear dynamical systems for the first time.
This paper develops the energy-phase method to deal with the high-dimensional non-autonomous nonlinear dynamical systems.In section 2, we formu late the problem and describe the geometrical structure of the phase space of the unperturbed systems, and we study the dynamics in the perturbed system, and derive the explicit formulas of the nth order energy difference function for the non-autonomous systems.In section 3, we apply our developed methods to a specific examp le: a two-mode truncation of parametrically excited, simply supported rectangular thin plate.And we visualize the ho moclinic tree and show by exp licit calcu lations how it breaks up under the effect of dissipation.We make a conclusion in section 4.

Generalized of the energy-phase method
Let us consider a two-degree-of -freedo m Hamiltonian system given by are periodic in t with the period Z S / 2 T , and 1) is an integrable Hamiltonian system on wh ich we make two structural assumptions: (H1) There exist converges at best conditionally.The improper integral makes sense if we approach the limits rf along the sequences of times there exist 1-pulse homoclinic orbits.
To formu late the above algorithm concisely, we introduce the energy difference function If equation ( 6) has transverse zero, then there exist Npulse homoclinic orbit.

Equations of motion of the buckled thin plate
We consider the simp ly supported at the four-edge rectangular thin plate whose edge lengths are a and b and thickness is h .The thin plate is subjected to its plane excitations.We establish a Cartesian coordinate system shown in Figure 2 and the coordinate Oxy is located at the middle surface of the thin plate.It is assumed that u , v and w represent the displacement of a point in the middle p lane of the thin p late in the x , y and z directions, respectively.The in -plate excitation of the thin plate may be expressed in the form .

Figure 2 The model of rectangular thin plate and the coordinate system
According to ref [11], the dimensionless equations of motion for the thin plate under parametric excitation are obtained as follows Using the normal form theory and the canonial transformation, equation ( 1) can be transformed to (10)

The unperturbed dynamics
When 0 H , it is noted that system ( 8) is an uncoupled two-degree-of-freedom nonlinear system since 0 I .Consider the first two decoupled equations , system (11) can exhib it the homoclinic bifurcation on the curve defined by

E
.When , the trivial zero solution may bifurcate into three solutions through pitchfork bifurcation, which are given by , respectively, where (12) Fro m the Jacobian matrix evaluated at the non-zero solutions, it is known that the singular point Thus, for all > @ 2 1 , I I I , system (11) has one hyperbolic saddle point 0 q which is connected to a pair of homoclin ic orb its , then the homoclin ic orb its can be obtained as follows Therefore, in the full five-dimensional phase space, the set defined by is a three dimensional invariant manifold.
Considering the unperturbed system of equation ( 8) restricted to M , we can calculate the resonant value and the phase shift J ' of the oscillations is defined as The n-th order energy-difference function for the dissipative case is given as follows: Co mputing (17) leads to the following exp ression for the dissipative energy difference function (18) where Defining a d issipative factor f d P such that d gives the relative measure of the dissipative effect with respect to the excitation amplitude.Hence the upper bound on the value of the dissipative factor is obtained as follows: For any s mall dissipative factor , we obtain an upper bound on the maximu m number of pulses , there are two transverse zeroes of the dissipative energy difference function in the interval , that is, and the sequence of sets Next we define the pulse sequence Further, we define the layer sequence  .In the left diagram, the horizontal line segments at each level N indicate that an infin ity of N -pulse orbits exist for all values of the phase shift in the interval below that line.And this diagram shows a fairly stable pulse distribution for lo wer pulses and increasing sensitivity to small changes in the parameters for higher pulses.In the right diagram, J ' can be regarded as an independent bifurcation parameter, and the resulting b ifurcat ion diagram is an infinite binary tree, which can be called the homoclinic t ree.The d iagram for the layer radii also has a secondary meaning: for fixed J ' , ^k N r gives the angular distance of the take-off curves of mult i-pulse orbits fro m the nearest center on the manifold.Figure 5 gives the pulse sequence and layer radius sequence as a function of the phase shift in the dissipative case when 100 N at the cross-section 0 I . The layer radius diagrams now do not refer to layers of periodic orbits, but their secondary meaning remains valid for the full dissipative system: they show the approximate angular distance of the take-off curves of multi-pulse orbits from the nearest sinks on the manifold.

Conclusion
In this paper, we have generalized the energy-phase method to deal with the non-autonomous nonlinear system and derive the explicit formu las of the energydifferent function.Applying our method to a two-mode approximation of parametrically excited, simply supported rectangular thin plate, we find the homoclinic tree both in the Hamiltonian case and the d issipative case are shown in Figure 4 and Figure 5.The pulse number diagrams show a variety of ways in wh ich N can change.And the layer radius diagrams gives the layers of periodic orbits, and the angular distance of the take-off cu rves of mu lti-pulse orbits fro m the nearest center on the man ifold in the Hamiltonian case.But the layer radius diagrams in the dissipative case just show the approximate angular distance of the take-off curves of mult i-pulse orbits fro m the nearest sinks on the manifold.

3 . 2 I 6 .
The perturbed dynamicsIt is noticed that the saddle point may persist under small perturbations, in particu lar, And restricting the system (8) on the manifold start by defining the energy sequence where

Figure 3 Figure 4
Figure 3The construction of the layer sequence However, all these sequence is finite due to (20).For any periodic orbit k N L J , the pulse nu mber is

Figure 4 Figure 5
Figure 4 The pulse sequence and the layer sequence as a function of a function of the phase shift in the Hamiltonian case