This paper is concerned with the stability analysis of nonlinear third order ordinary differential equations of the form . 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 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.
In 1892, Lyapunov [
Consider the system
where x denotes an n-dimensional vector and
The trivial solution
Suppose there is a function V which is positive definite along every trajectory of (1), and is such that the total derivative
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! [
1) Eigenvalue analysis concept does not hold good for nonlinear systems [
2) Nonlinear systems can have multiple equilibrium points and limit cycles [
3) Stability behaviour of nonlinear systems need not always be global (unlike linear
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.
Given Equation (2), multiply by
We observe that;
Hence substituting in (4) we get;
Integrating with respect to t,
The required Hamiltonian is;
1) Any dynamical system of the form
that
2) The function
date in the stability analysis of many conservative systems.
3) A concrete example of a conservative system is the simple pendulum [
We consider nonlinear time-invariant system
An equilibrium solution
1) stable if, given any
2) uniformly stable if, for every
3) unstable if it is not stable.
4) asymptotically stable if there exists a
5) The system is globally asymptotically stable (G.A.S.) if for every trajectory
6) The system is locally asymptotically stable (L.A.S.) near or at
Ogundare [
which is equivalent to the system
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
Solving the system we have,
By setting C = 1, we obtain
with
We now find the time derivative of V:
Equation (13) is positive definite with
Exploiting the similarities between linear and nonlinear systems, we shall construct a Lyapunov function for the stability of third order nonlinear differential equation of the form;
Equation (15) is equivalent to the system
Taking into account the similarities between the linear and the nonlinear systems, we can see a close comparison with Equations (16) and (7), that
We use Equation (13) as our trial Lyapunov function, given by
We also assume that
This implies that
Since
Hence V is positive definite.
Also, the time derivative of V along the solution path of (16)
Clearly
Therefore, the Lyapunov function;
is an appropriate Lyapunov function for the system (16).
We have seen that the Lyapunov function candidate constructed in this project is a good tool in the stability analysis of dynamical systems. Without the need to solve the systems of differential equations involved, we were able to obtain the qualitative behaviour of the systems near their equilibrium points.
Okereke, R.N. (2016) Lyapunov Stability Analysis of Certain Third Order Nonlinear Differential Equations. Applied Mathematics, 7, 1971- 1977. http://dx.doi.org/10.4236/am.2016.716161