ABSTRACT

This paper presents a comprehensive stability analysis of the dynamics of the damped cubic-quintic Duffing oscillator. We employ the derivative expansion method to investigate the slightly damped cubic-quintic Duffing oscillator obtaining a uniformly valid solution. We obtain a uniformly valid solution of the un-damped cubic-quintic Duffing oscillator as a special case of our solution. A phase plane analysis of the damped cubic-quintic Duffing oscillator is undertaken showing some chaotic dynamics which sends a signal that the oscillator may be useful as model for prediction of earthquake occurrence.

1. Introduction

Most real life problems are nonlinear in nature. This has made the study of nonlinear systems which are very complex an important area of study and research. The Duffing oscillator is one of such important nonlinear system.

System (2) below describes the motion of the cubic Duffing oscillator which can be used to model conservative double well oscillators which can occur in magnetoelastic mechanical systems [1]. A good and illustrating example of such system was described in [2]. The cubic Duffing equation can as well be used to model the nonlinear spring-mass system [3,4], as well as the motion of a classical particle in a double well potential [5]. System (2) were proposed by Correig in [6] as a model of microseism time series and have been used in [7] to model the prediction of earthquake occurrence. It was also used to model the transverse oscillation of nonlinear beams in [8].

Generally, the Duffing oscillator can be described by the following equation of motion:

(1)

where, are arbitrary positive or negative constants

Is the Hamiltonian, is angular frequency and is the amplitude of the harmonic external periodic force.

If we set in (1), we obtain,

(2)

and the Hamiltonian becomes

Generally, the damped and forced cubic-quintic Duffing oscillator with random noise obtained by setting in (1) is given by the equation

(3)

where,. Here is the damping coefficient, is the proper or resonant frequency, while β and µ are the coefficient of nonlinearity. is the random noise.

We can write (3) as a system in the form,

(4)

where,

is a tri-stable potential or a triple well potential.

Setting in Equation (4), then we get

(5)

where implies, from (4),

(6)

The stability matrix of the system (4) is given by,

(7)

The corresponding characteristic equation given by det. is

(8)

2. Perturbative Analysis

Many applied mathematicians have applied the different versions of the multi-scale method which includes the derivative expansion method to a wide variety of problems in physics, engineering and applied mathematics obtaining valid, useful and good results as in [9-14]. The versatility of the derivative expansion method is exhibited where other multiple-scales approach for finding approximate solutions to nonlinear differential equations fails as noted in [15].

Here, we use the derivative expansion method to obtain three-term uniformly valid approximate solution of the slightly damped and forced cubic-quintic Duffing equation.

Presently, it is not possible to obtain an exact solution to nonlinear differential equations like our cubic-quintic Duffing oscillator. But, over the years, applied mathematicians have developed methods used successfully to obtain good approximate solutions to these nonlinear differential equations. Among these methods are the perturbation methods like, the method of averaging with its versions, Linstedt-Poincare’s method, Lighthill technique, matched asymptotic methods, multi-scale method with its versions, homotopy perturbation method [16-18], nonperturbative methods [19,20] like, the global error minimization method (GEM) [21], the variational approach [22,23], variational iteration method [24,25], homotopy analysis method [26] and the energy balancing method [27], cubication method [28], and Newton-harmonic balancing method [29].

The method of averaging which is a powerful perturbation method in which solutions of autonomous dynamical systems can be used to approximate solutions of complicated time-varying dynamical systems as mentioned in [30], as well as the Linstedt-Poincare method has been used in to obtain a two-term uniformly valid approximate solution of the un-damped and slightly forced cubic-quintic Duffing equation.

Consider the slightly damped and forced cubic-quintic Duffing equation given by,

(9)

where, is the natural frequency, ε is a very small term,

We assume that the solution to (9) can be written in the form,

(10)

where

Let

(11)

so that,.

Substituting (10) and (11) into (9) and equating the coefficient of the powers of epsilon to zero, we obtain,

(12)

(13)

(14)

where,

the linear operator The solution to (12) is given as,

(15)

where is a complex function and is its conjugate.

Substituting (15) into (13) and eliminating secular terms, we obtain,

(16)

and,

(17)

where CC denotes complex conjugate.

Solving (16) by letting

with real we obtain

and

(18)

Substituting (15) and (17) into (14) and eliminating secular terms, we obtain,

(19)

(20)

where

(21)

Employing (19) in (17) and solving for the third uniformly valid term, we obtain,

(22)

where,

Note that we have used the fact that the complementary function arising from (14) is zero for a uniformly valid solution.

Solving (20) by letting

with real and and using (18), we obtain,

and

(23)

where k and are constants and

.

Then using (18) and (23), we obtain,

(24)

(25)

and U₁ in (17) becomes

Using (15), (21), (22) with (23), (24) and (25) the uniformly valid three-term solution to (10) is given by,

(26)

where ,the quasi-frequency is given by:

(27)

a constant.

For the two-term approximate solution of the undamped and forced cubic-quintic Duffing equation, we set in (26) to obtain,

          

 (28)  

Equation (28) is a valid approximate solution of the un-damped and forced cubic-quintic Duffing oscillator.

3. Stability Analysis

The unforced and damped cubic-quintic Duffing oscillator from (3) is given by,

(29)

From (29), we obtain the autonomous dynamical system,

From the condition, we obtain which gives us,

(30)

Writing (30) as,

(31)

We obtain the roots of (31) as,

(32)

Then the equation satisfied by the eigenvalues of our systems stability matrix is,

(33)

where denotes or the coordinate of an equilibrium point.

Whether our eigenvalues will be complex, real or imaginary will be determined by the values of

With. For this we consider the following cases:

1): for this case

This contradicts our assumption that det. and it also implies that are all zero.

2): for this case.

This corresponds to critical points that are centres for which stability is ensured.

3): for this case which corresponds to saddles giving rise to instability [31].

With

.

Then considering the cases below we have:

1) giving the values which goes contrary to our assumption of det.. It also implies that. are all zero.

2): for this case

Considering the discriminant for this case, we have the three cases viz,.

For the case

which corresponds to spirals and asymptotic stability [32].

The case gives values of the form

which correspond to nodes resulting in asymptotic stability if and to saddles and consequent instability if

The case gives

For this case we have saddles and hence instability 3) lead us to have

Considering the discriminant we consider the three possible cases as follows:

a): this case gives

and we have spirals and consequently asymptotic stability (see [31]);

b) gives

and this leads us to existence of nodes and instability if or saddles and instability if

c) is the case for which which yield centres and stability.

We now consider the stability of the dynamics for a few choices of employing Equations (32) and (33). In doing this we consider two segments given by. For each of these segments we treat five cases. Table 1 summarizes the dynamics for the few choices of parameters we used.

4. Discussion of Results

Figure 1, is the phase plot showing the behavior of the oscillator when and. It is periodic. We observe asymptotically stable spirals at (0, 0) and

which are the three equilibrium points observed. This situation is seen in the cubic Duffing oscillator.

The phase plot in Figure 2 shows the dynamics of the oscillator when and. Five equilibrium points are observed. There exists an asymptotically stable spiral at (0, 0) and unstable saddles at and. This situation is not observed in the cubic Duffing oscillator. The five equilibrium points generated is as a result of the presence of the quintic term.

The phase plot in Figure 3 exhibits the behavior of the oscillator when and. Five equilibrium points are observed in this phase plot. There exist asymptotically stable spirals at (0, 0) and while unstable saddles were obtained at. This situation cannot occur in the cubic Duffing oscillator. As noted above, it is the addition of the quintic term that allows such situations to occur.

The phase plot in Figure 4 exhibits the behavior of the oscillator when and. it is quasi-periodic. Five equilibrium points are observed. There are asymptotically stable spirals at and while a saddle exists at (0, 0). As mentioned above, this situation is possible due to the addition of the quintic term. This situation also cannot occur in the cubic Duffing oscillator.

The phase plot in Figure 5 shows the dynamics of the oscillator when and. It is quasi-periodic. Five equilibrium points were also observed. Saddles were obtained at (0,0) and while asymptotically stable spirals are obtained at. This situation also cannot occur in the cubic Duffing oscillator, the situation is made possible by the addition of the quintic term.

Table 1. Stability of the dynamics of the oscillator for few choices of parameter.

The phase plots in Figures 6-10 were obtained for the same values of respectively but now with, the same analysis hold for these cases. It is very important to note that in the phase plots obtained for, the phase lines tend to converge (move towards) to the equilibrium points while for, the phase lines diverge (move away) from the equilibrium points to infinity. This development is in harmony with the solution we obtained in (32) for the damped and forced cubic-quintic Duffing oscillator where, setting, we found out that the exponential function depicting the damping grows larger and tends to infinity. Setting, we found out that the exponential function becomes smaller and tends to zero. These results are in line

with observations noted in [33-35] where negative damping effects were observed.

Figure 11 depicts the behavior of the frequency of the unforced and un-damped cubic-quintic Duffing equation obtained using Linstedt-Poincare method (LPM), derivative-expansion method (DEM), homotopy analysis method (HAM) and the method of averaging technique (MOA) for increasing amplitude. Among the three perturbation techniques used, the derivative expansion method agreed most with the homotopy analysis method for small parameters.

In Figures 12 and 13, one observes the dynamics of the damped and forced cubic-quintic Duffing equation as time increases, where we have taken .

Figure 14 shows the periodic displacement of the