Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations

This paper is concerned with the stability analysis of nonlinear third order ordinary differential equations of the form ( ) 0 x ax g x cx + + + =    . We construct a suitable Lyapunov function for this purpose and show that it guarantees asymptotic stability. Our approach is to first consider the linear version of the above ODE, by taking ( ) g x bx =   and study its Lyapunov stability. Exploiting the similarities between linear and nonlinear ODE, we construct a Lyapunov function for the stability analysis of the given nonlinear differential equation.


Introduction
In 1892, Lyapunov [1] proposed a fundamental method for studying the problem of stability by constructing functions known as Lyapunov functions.This function is often represented as ( ) , V t x defined in some region or the whole state phase that contains the unperturbed solution x = 0 for all t > 0 and which together with its derivative ( )  satisfy some sign definiteness.The following definitions of stability were given by Lyapunov.

Definition (Lyapunov)
Consider the system ( ) ( ) where x denotes an n-dimensional vector and ( ) ; , x t x t be a solution of the Equation (1) through ( ) 0 0 , x t then the trivial solution ( ) 0 0 , 0 x x t = of the system (1) is said to be stable at 0 t t = , provided that for arbitrary positive 0 ε > , there exist a ( ) is satisfied for all 0 t t ≥ .

Definition (Lyapunov)
The trivial solution ( ) 0 0 ; , x t x t of the system (1) is said to be asymptotically stable if it is stable, and for each 0 0 t > , there is an ; , 0 x t x t → as t → ∞ .If in addition all solutions tend to zero, then the trivial solution is asymptotically stable in the large.

Lyapunov's Theorem on Stability (Lyapunov)
Suppose there is a function V which is positive definite along every trajectory of (1), and is such that the total derivative V  is semi definite of opposite sign (or identically zero) along the trajectory of (1).Then the perturbed motion is stable.If a function V exists with these properties and admits an infinitely small upper bound, and if V  is definite (with sign opposite of V), it can be shown further that every perturbed trajectory which is sufficiently close to the unperturbed motion 0 x = approaches the latter asymptotically.

Remark
1) The basis of Lyapunov theory in simple terms is that; if the total energy is dissipated, then the system must be stable.
2) The main advantage of this approach is that; by looking at how an energy-like function V (Lyapunov function) changes over time, we might conclude that a system is stable or asymptotically stable without solving the differential equation.
3) The disadvantage of this approach is that; finding a Lyapunov function may not be so easy![2].

Motivation
1) Eigenvalue analysis concept does not hold good for nonlinear systems [1] [3]. 2) Nonlinear systems can have multiple equilibrium points and limit cycles [4].

How Energy Is Associated with Dynamical Systems
We illustrate here how we can derive the Hamiltonian for a dynamical system of the form The Hamiltonian of a system is the sum of its kinetic (T) and potential energies (V), i.e.

Remark
1) Any dynamical system of the form is conservative [7], i.e. the total energy of the system is conserved, and this implies that there exists a function H such that d 0 d

H t
= , where H is the Hamiltonian of the system.
2) The function ( ) is often used as a Lyapunov function candidate in the stability analysis of many conservative systems.3) A concrete example of a conservative system is the simple pendulum [8].

Stability Definitions
We consider nonlinear time-invariant system ( ) x ∈  is an equilibrium point of the system if ( ) 0 e f x = .We remark that e x is an equilibrium point if and only if ( ) e x t x = is a trajectory.Figure 1 illustrates schematically the concept of stability and asymptotic stability with respect to an equilibrium point n e x ∈  .Their definitions follow.

Definition
An equilibrium solution e x x = of ( ) is said to be: 1) stable if, given any 0 ε > and any 0 0 t ≥ , there exists a ( ) 2) uniformly stable if, for every 0 ε > , there exits ( ) , independent of 0 t , such that (3.3.1) is satisfied for all 0 0 t ≥ , 3) unstable if it is not stable.

Construction of Lyapunov Function for Third Order ODE
Ogundare [9] constructed a Lyapunov function for a second order linear system of ordinary differential equation using a quadratic form.We shall adapt his method with slightly more simplified assumptions to construct a Lyapunov function for a third order linear ODE of the form which is equivalent to the system x y y z z az by cx where a, b, c are all positive constants.The required quadratic form in this case is given as where A, B, C, D, E, and F are constants to be determined.Differentiating Equation (8) with respect to the system (7) we have Setting the coefficients of , , , , xy xz yz x z to be zero and coefficient of 2 y greater than zero in Equation ( 9), we obtain Solving the system we have, ( ) By setting C = 1, we obtain ( )

V
these values of the constants guarantee the positive definiteness of V and negative definiteness of its derivative.So, substituting (12) into Equation (8), gives acxy b a yz z az by cx cy cxz az ay az by cx acxy byz a yz az byz cxz cy cxz az a yz aby acxy cy aby c ab y ab c y