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 earth- quake occurrence.

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 [

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

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,

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

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,

where,

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

Setting in Equation (4), then we get

where implies, from (4),

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

The corresponding characteristic equation given by det. is

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 [

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) [

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 [

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

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

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

where

Let

so that,.

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

where,

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

where is a complex function and is its conjugate.

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

and,

where CC denotes complex conjugate.

Solving (16) by letting

with real we obtain

and

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

where

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

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

where k and are constants and

.

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

and U_{1} in (17) becomes

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

where ,the quasi-frequency is given by:

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.

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

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

From the condition, we obtain which gives us,

Writing (30) as,

We obtain the roots of (31) as,

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

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 [

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 [

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 [

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.

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

The phase plot in

The phase plot in

The phase plot in

The phase plot in

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.

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 .