Stability Analysis of Periodic Solutions of Some Duffing ’ s Equations

In this paper, some stability results were reviewed. A suitable and complete Lyapunov function for the hard spring model was constructed using the Cartwright method. This approach was compared with the existing results which confirmed a superior global stability result. Our contribution relies on its application to high damping door constructions. (2010 Mathematics Subject Classification: 34B15, 34C15, 34C25, 34K13.)


Introduction
In real life, most problems that occur are non-linear in nature and may not have analytic solutions except by approximations or simulations and so trying to find an explicit solution may in general be complicated and sometimes impossible.Duffing's equation is a second order non-linear differential equation used to model such problems of non-linear nature [1].In particular, it is used to model damped and driven oscillators, for example, modelling of the brain [2], in prediction of earthquake occurrences [3], signal processing [4] and crash analysis [5].In [6], differential equation which describes a non-linear oscillation was first introduced by Duffing's with cubic stiffness constant.The general form of Duffing's equation is: where ( ) p t is continuous and 2π-periodic in t ∈  .
The study of periodic solution of Duffing's equation is like that of the study of classical Hamiltonian equation of motion which is characterized by multiplicity of periodic solution.Various techniques for investigating the stability of periodic solutions of Duffing's equation have been reported, (see [7] [8] [9] [10] [11]).
Our approach to study stability of periodic solutions is by the use of the Lyapunov direct method.Many researchers have obtained useful and valid results using the Lyapunov second method (direct method) for stability analysis and construction of appropriate Lyapunov functions for other Duffing equation but the use and construction of Lyapunov function using Cartwright method are rare in literature, for instance [12] [13] [14] [15] [16].The application of this method is in constructing a scalar function and its derivatives with peculiar characterizations.When these characterizations are satisfied, the stability behaviour of the system is solved.The difficulties in the construction of suitable Lyapunov functions in nonlinear systems have attracted the attention of many researchers and have been summarized in [17] [18] [19] that discussed the general approach to the stability study of periodic solution which is related to the classical Lyapunov theorem based on linear approximations.This reduces the stability study of periodic solutions to the stability of the system linearized at the periodic motion.
Since linearized systems contain periodic coefficients, the theory of parametric resonance can be applied.Such approach with the analysis of Floquet multipler is used in [9] [20] [21].The other traditional approach to the study of stability of periodic solutions is the approximate average and multiple scales method, which reduces original time dependent dyamical systems to autonomous system.In this case stability study is the analysis of fixed point, (see [22]).
Our construction of suitable Lyapunov function rests squarely on the approach of [23].Others who used this approach are [16] [24].Very few nonlinear systems can be solved explicitly and so we must rely on numerical schemes to approximate the solutions, (see [25] [26] [27] [28]).This paper is motivated by studying [11] [29] where stability analysis of the unforced and damped cubic-quintic Duffing oscillator of the form was carried out.Using Equation (1.2) the existence of a three term valid solution was obtained using derivative expansion method.The derivative expansion method is one of the perturbative methods which require the existence of a small parameter and is therefore not valid in principle for the Duffing equation in which the nonlinearity is large.Its stability analysis of the equilibrium point was carried out using the eigenvalue approach.Secondly, our motivation was augmented by reading the works of [10] where new criteria for existence and asymptotic stability of periodic solutions of a Duffing equation of the form ( ) were derived.This approach exposed by [10] is useful when the non-linearity does not admit the decomposition of ( ) ( ) ( ) ( ) ( ) where , , a b c are real constants and [ ] : 0, 2π h →  N is continuous.In Equa- tion (1.4), a is the stiffness constant, c is the coefficient of viscous damping and

Preliminaries
Definition 2.1.(Properties of Lyapunov Function) The Lyapunov function has the following properties: a) Continuity: ( ) , V t X is continuous and single valued under the condition t T ≥ and i x H < and ( ) , representing the total derivative with respect to t is negative definite.

Definition 2.2. (Complete Lyapunov function)
A Lyapunov function V defined as : where c is any positive constant and X given by ( ) Theorem 2.3.Consider a system of differential equations ( ) be an equilibrium point of Equation (1.6).
Theorem 2.6.Suppose all conditions for asymptotic stability are satisfied.In addition to it, suppose there exists constants 1 2 3 , , , ; Moreover, if this conditions hold globally, then the origin 0 X = is globally exponentially stable.

Construction of Lyapunov Function for the Duffing Equation Using Cartwright Method
We adapt the method of construction of Lyapunov function used in [23] and extend it to the second order non-linear differential equation of the Duffing type of the form (3.1).In the sequel, [34] asserted that Lyapunov functions are vital in determining stability, instability, boundedness and periodicity of ordinary differential.Considering the Duffing equation of the form ( ) The equivalent systems of Equation (1.7) is ( ) where ( ) and ( ) 0 Writing the equivalent systems of Equation (1.7) in compact form, we have where ( ) The method discussed here is based on the fact that the matrix A has all its ei-genvalues as negative real parts.Then from the general theory which corresponds to any positive quadratic form ( ) U x , there exists another positive defi- nite quadratic form ( ) Choosing the most general quadratic form of order two and picking the coefficient in the quadratic form to satisfy Equation (1.10) along the solution paths of Equation (1.10) we assume V to be defined by

V Axx Byy K xy yx Axy B cy ax h x Ky Kx cy ax h x
Axy Bcy Byax Byh x Ky Kcxy Kax Kxh x Simplifying the coefficients we have To make V  negative definite, we equate the coefficient of mixed variable to zero and the coefficients of 2 x and 2 y to any positive constant (say δ ) as outlined in Table 1 0 From Equation (3) above Then substituting the value of K into Equation ( 2) we obtain Bc a Substituting for K and B in (1) we have ( ) which by further simplification gives that ( ) Substituting for the values of the constant A, B, K in Equation (1.11) gives Using Equation (1.19) and the fact that 0 V <  , the equilibrium point is asymptotically stable.

Stability Analysis of Periodic Solution of Duffing Equation Using the Eigenvalue Approach
Considering the Duffing equation of the form (1.7), the first equivalent systems of (1.7) is given by For the unforced case, Equation (1.19) is reduced to The matrix equation for (1.21) can be written as Examining the stability at the point (0,0) gives ( ) where the coefficient of 2 x is 6.
At the origin, ( ) ( ) For this, we consider the following cases: 1) When 0 g = , 1,2 0 λ = and this implies that , , b x a are all zero; 2) When 0 g > , 1,2 i g λ = ± which corresponds to critical points that are centres for which stability is ensured; which corresponds to saddles giving rise to instability [11].
With 0 δ > , ( ) For this, we consider the following cases: ) For the discriminate, we have the following cases: shows that the roots are imaginary which to lead to spiral and asymptotic stability.
shows that the roots are real which leads to saddles and instability. When shows that the roots are real corresponding to saddles and instability.When 0 g < , ( ) For the discriminate, we consider the following cases: 1) When 2 4 0 g δ + < , ( )

Numerical Solution of Duffing's Equation Using Mathcad
Derivative function.

Stability Analysis of Duffing Equation under Oyesanya and Nwamba (2013)
The unforced and damped cubic-quintic Duffing oscillator is given by From Equation (1.32) we obtain the autonomous dynamical system From the condition 0 u =  we obtain 0 y = which gives us Equation (1.34) can be written as We obtain the roots of (1.35) as Then the equation satisfied by the eigenvalues of our systems stability matrix is where u denotes 1 2 3 u u u and 4 u or the u-coordinate of an equilibrium point.
Whether our eigenvalues will be complex, real or imaginary will be determined by the values of δ and For this we consider the following cases: 1) 0 g = : for this case 1,2 0. λ = This contradicts our assumption that det 0 J ≠ and it also implies that , , α β µ are all zero.
2) 0 g > : for this case 1,2 .i g λ = ± This corresponds to critical points that are centers for which stability is ensured.
2) 0 g > : for this case Considering the discriminant fot this case, we have three cases.gives 1,2 g λ = ± .For this case we have saddles and hence insta- bility.
3) 0 g < leads us to have Considering the discriminate we consider the three cases as follows: Open Journal of Applied Sciences a) 2 4 0 g δ + < : this case gives  which yield centers and stability.We now consider the stability analysis of the dynamics for a few choices of , , α β µ and δ by using employing Equations (1.35) and (1.36).These are il- lustrated in Table 3.

Stability Analysis of Torres (2004)
We consider the Duffing's equation of the form ( ) is a 1 L -caratheodory function such that the partial derivative x g exists and it is: 1 L -caratheodory.For a given 1 p ≤ ≤ +∞ , let us define the set with * , p k defined.Let ϕ be a 2π periodic solution of Equation (1.37), we Table 3. Stability analysis of the dynamics of the oscillator for few choices of parameter α, β and μ.For such an a, the operator L as defined in maximum principle is inversely positive.By defining the operator.
[ ] ( ) The solution ϕ can be seen as a fixed point of the operator H. Next, we recall that an isolated 2π-periodic solution has a well-defined index ( ) γ ϕ and ( ) 1 γ ϕ ≤ .

Discussion
. This shows that the phase is revolving round it.At (0.468) the point was saddle showing that it is unstable.In this case, there is an increase in oscillation which is as a result of decrease in the damping coefficient.That is decrease in damping leads to increase in oscillation.Figure 6: The phase portrait of Duffing's equation was obtained.We observed that the phase line was depicting a center of non-stable node. .In this case, there is a decrease in the maximum displacement from the origin which leads to decrease in oscillation.The decrease in oscillation is as a result of increase in the damping coefficient.That is increase in damping leads to decrease in oscillation.

Conclusion
The study of the stability analysis of periodic solution using Lyapunov direct Open Journal of Applied Sciences . Analysis of the stability of the fixed points can be done by linearizing Equation (1.20) which gives The coefficient of β shows that the Duffing equation is highly damped representing a hard spring.This hard spring is represented in Equation(1.21) Number of solution values on [ ] 0

34 )
Open Journal of Applied Sciences

Figure 1
Figure 1 is the MATCAD showing the behavior of the Duffing's equation when 0.1, 0.2 a b = = and 0.1 c = is periodic.We observe asymptotically stable at both saddle and spiral point i.e.

Figure 3 :
Figure 3: the MATCAD shows the dynamics of the Duffing's equation when 0.01, 0.02, 0.03 a b c = = = .Three equilibrium points was observed.Saddles were obtained at ( ) 0.02, 0.066 − showing that the phases are toward the origin.Figure 4: the MATCAD were obtained for the values of 0.2, 0.2, 0.03 a b c = − = =.In this case it is important to note that the phase line

Figure 4 :
the MATCAD were obtained for the values of 0In this case it is important to note that the phase line tend to converge toward the equilibrium points.

Figure 5 :
The trajectory profile of Duffing's equation were obtained for the values 0

Figure 7 :
The trajectory profile of Duffing's equation were obtained for the values 0

Figure 8 :
Figure 8: In this case, the phase portrait was depicting the center as an unstable node.

Figure 2 .
Figure 2. Phase portrait of Duffing's equation showing asymptotic stability of solution as a spiral sink.

Figure 4 .
Figure 4. Phase portrait of Duffing's equation depicting asymptotic stability of solution as a spiral sink.

Figure 6 .
Figure 6.Phase portrait of Duffing's equation depicting a centre of a non-stable node.

Figure 8 .
Figure 8. Phase portrait depicting the centre as an unstable node (parameters).

Table 1 .
A table showing terms and coefficients of Equation (1.12).

Table 2 .
The stability analysis and numerical solutions of duffing's equation at different values a, b, c.
We consider the stability analysis of Duffing's equation for different values of a, b, c, as illustrated in Table2.