We analyze the dynamics of the forced singularly perturbed differential equations of Duffing’s type with a potential that is bounded from above. We explain the appearance of the large frequency nonlinear oscillations of the solutions. It is shown that the frequency can be controlled by a small parameter at the highest derivative.

1. Introduction

Duffing’s equation is regarded as one of the most important differential equations because it appears in various physical and engineering problems. For example, the periodically forced Duffing oscillator exhibits a wide variety of interesting phenomena which are fundamental to the behavior of nonlinear dynamical systems, such as regular and chaotic motions (see, e.g., [15] and the references therein; we also refer to the classical book of Nayfeh and Mook [6]). In this context, usually two-well potential of an unperturbed system was considered (; see also [79]) by using analytical methods and numerical simulations.

In this case, the undamped () and unperturbed () Duffing’s oscillator can basically exhibit two distinct types of steady-state oscillations, namely, (i) in-well, small orbit dynamics, where the system state remains within the potential well centred at a stable equilibrium point (center); (ii) cross-well, large orbit dynamics, whose trajectories surround the three equilibrium points (saddle between two centers). In both cases, under periodic external excitation, a chaotic motion can be observed when the control parameters are changed.

On the contrary, in this paper we consider that and a potential tends to for , so the object can escape to infinity because of the bounded from above potential. There exist several atomic or subatomic situations in quantum physics where the total energy governing the particles contains an approximately square-well potential which is bounded from above; see, for example, [10] and discussion in [11]. For example, recently it has been found that the meson spectroscopy is better described by “confining” potential which is bounded from above; for details and references, see [10].

This paper concentrates on the mathematical aspects of systems with a potential that is bounded from above; more concretely, we focus our attention on the existence of nonlinear oscillations in the context of saddle-center bifurcation in the dynamical system describing the singularly perturbed forced oscillator of Duffing’s type with a nonlinear restoring and a nonperiodic external driving force or rewriting to an equivalent set of the three first-order autonomous equations: with potential with two-barrier structure. Here and are the functions on the interval is positive and is a function on .

We show that the singular perturbation parameter play role modeling tool for the frequency control of the nonlinear oscillations arising in these systems (relationship (33)). Finally we prove that under some assumptions the solutions of (2) will rapidly oscillate, with the frequency of the oscillations increasing unboundedly as .

System (3) is an example of a singularly perturbed system, because in the limit , it does not reduce to a differential equation of the same type, but to an algebraic-differential reduced system:

Another way to study the singular limit is by introducing the new independent variable which transforms (3) to the system Taking the limit , we obtain the so-called associated system ([12]) in which plays the role of a parameter.

Both scalings agree on the level of phase space structure when but offer very different perspectives since they differ radically in the limit when . The main goal of singular perturbation theory is to use these limits to understand structure in the full system when .

The critical manifold is defined as a solution of the reduced system; that is, which corresponds to a set of equilibria for the associated system (6), (7), and (8).

2. Saddle-Center Bifurcations of Associated System

We assume the following. (A1) The critical manifold is S-shaped curve with two folds; that is, it can be written in the form , and the function has precisely two critical points, one nondegenerate minimum and one nondegenerate maximum ; let . Thus, the critical manifold can be broken up into three pieces , and , separated by the minimum and maximum (Figure 1). These three pieces are defined as follows:(A2)Consider for .(A3)Consider for every and for every .

Due to the assumption (A3), the situation considered here substantially differs from the situation in [13], where two pieces of critical manifold, namely, and , are not normally hyperbolic, and large orbit oscillations that encircle all the three pieces of critical manifold were studied (for the definition of a normal hyperbolicity of critical manifold see, e.g., [12]). In this paper, the pieces and of the critical manifold are normally hyperbolic, and thus the system under consideration allows another type of nonlinear oscillations, namely, small orbit oscillations around the middle piece of critical manifold . For comparison, see Figure 3(a) and Figure 5.

Let ,  . Denote by

The equations have three solutions for if and one if and . Thus the associated system (6), (7), has three equilibria (two saddles, one center) for and one equilibrium (saddle) for and ; the eigenvalues of the jacobian are Thus, the pieces and of critical manifold are the normally hyperbolic manifolds. The points and of are the cusps of the codimension two ([14, 15]), and the corresponding bifurcation is known as saddle-center bifurcation ([16]) (Figure 2).

More precisely, at the point , the birth of the saddle and the center occurs. At , the left side center and the saddle coalesce. There is a unique , such that between hyperbolic points (saddles) there is a heteroclinic connection. The homoclinic loop of one hyperbolic point surrounds the corresponding elliptic (center) one for every .

We use the level surfaces of the energy function , to characterize the trajectories of (3). These surfaces in -space are defined by extending it as long as remains real. On the intervals and , there is a motion across a single potential barrier, and on the interval , there is double barrier with a well in between.

The derivative of along any solution path of (3) is

The main objective of this paper is to prove the strongly nonlinear oscillations on the interval as possible scenario of behavior of the solutions for problem (2). For this reason, we rewrite the differential equation (2) in the system form Then, we make a change of variables from rectangular coordinates to the dynamic polar coordinates centered at defined by the equations and let us consider the system (17) on the interval in these new coordinates. The function acting in the first polar transform equation corresponds to the middle piece of critical manifold . At first, we derive the differential equation for polar angle , which is crucial for an analysis of nonlinear oscillations in this system. Dividing formally the second transform equation by the first, we get Differentiating this equation with respect to , we consecutively have

Finally, using (17) and (18), we obtain the following differential equation for where the radius is as follows:

Now let be a compact subset of . On the periodic orbits (for fixed ), we define the minimal radius: where and () are the solutions of the equation lying on the periodic orbit.

Obviously, for and for .

Let be a positive function such that on .

Now we make the following additional assumption. (A4) The total energy of motion described by (2) satisfies on a compact subset of .

If a total energy of motion described by (2) satisfies the assumption (A4) on every compact subset of , then for and for , and .

We precede the main result on the existence of nonlinear oscillations of the solutions for (2) on the interval by important example.

Example 1. Consider Duffing’s oscillator with linear excitation for subject to the initial conditions on the interval .
In our case , , , , , , , , , and it is not difficult to check that the assumptions (A1)–(A3) hold. Numerical results obtained from (27) subject to the initial conditions (28) using the software package MATLAB 7 are shown in Figure 3(a). These oscillations are very sensitive on the value of singular perturbation parameter . Figure 3(b) shows the solution of (27), (28) for .
In order to facilitate the understanding of the qualitative behaviors of this dynamical system, we draw the -phase portraits at the relevant fixed value of time (Figure 4). For comparison, Figure 5 shows the solution of initial value problem with twin-( or single-) well potential for on the interval . This type of problems has been studied in [13].
Now we will analyze the solution of (27), (28) after the time . The total energy (14) for (27) is and for its derivative along the solution given by (16), we obtain Due to the oscillations around in left neighborhood of between and , that is, , the total energy decreases (dissipation) in right neighborhood of , As it follows from the shape of potential and the phase portrait for for and .

3. Analysis of Solutions Lying on the Periodic Orbits: Main Result

Now we formulate the theorem on nonlinear oscillations of solutions of (2) for the motion with total energy satisfying the assumption (A4). Moreover, we show that the parameter plays role modeling tool for the frequency control of the nonlinear oscillations.

Denote by the spacing between two successive zero numbers of the function on , where is a solution of (2).

Theorem 2. Under the assumptions (A1)–(A4), the solutions of problem (2) oscillate on the compact set between and and where and are the positive constants independent of the singular perturbation parameter .

Proof. To obtain the oscillations and the estimate of their frequencies, we analyze the differential equation (21); that is, Taking into consideration the fact that independently of parameter due to the assumption (A4), we can estimate that for . Further for and , we have . Thus for sufficiently small values of the singular perturbation parameter , say, , there exist the positive constants , and where , and are the roots of the equation lying on the periodic orbit.
Putting (35) into the definitions of constants , and , we obtain that for sufficiently small value of upper bound of singular perturbation parameter .
Further, if , then and conversely
Thus we have the inequality Integrating this inequality with respect to the variable between two successive zeros ( and ) of , we obtain immediately the lower and upper bound of their spacing . Indeed, Hence,

4. Conclusion

The frequency of nonlinear oscillations of Duffing’s type equations arising via saddle-center bifurcation in associated system may be controlled by the singular perturbation parameter . These oscillations are very sensitive on the initial conditions.