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The global phase portrait describes the qualitative behaviour of the solution set for all time. In general, this is as close as we can get to solving nonlinear systems. The question of particular interest is: For what parameter values does the global phase portrait of a dynamical system change its qualitative structure? In this paper, we attempt to answer the above question specifically for the case of certain third order nonlinear differential equations of the form . The linear case where is also considered. Our phase portrait analysis shows that under certain conditions on the coefficients as well as the function , we have asymptotic stability of solutions.

A line connecting the plotted points in their chronological order shows temporal evolution more clearly on the graph. The complete line on the graph (i.e. the sequence of measured values or list of successive iterates plotted on a phase space graph) describes a time path or trajectory [

An orbit for a system usually indicates that the dynamical system under consideration is conservative. We also note that each plotted point along any trajectory has evolved directly from the preceding point. As we plot each successive point in phase space, the plotted points migrate around. Orbits and trajectories therefore reflect the movement or evolution of the dynamical system. Thus, an orbit or trajectory moves around in the phase space with time. The trajectory is a neat, concise geometric picture that describes part of the system’s history. When drawn on a graph, a trajectory must not always be smooth; instead, it can zigzag all over the phase space, mostly for discrete data [

The phase space plot is a world that shows the trajectory and its development. Depending on various factors, different trajectories can evolve for the same system. The phase space plot and such a family of trajectories together are a phase space portrait, phase portrait, or phase diagram.

A phase space with plotted trajectories ideally shows the complete set of all possible states that a dynamical system can ever be in.

We next describe the notion of the flow of a system of differential equations. We begin with the linear system

The solution to the initial value problem associated with (1) is given by

The set of mappings

The mapping

1)

2)

3)

For the nonlinear system

we define the flow

by

Let E be an open subset of

is called a flow of the differential Equation (2).

1) We can think of the initial point as being fixed and let

2) If we think of the differential Equation (2) as describing the motion of a fluid, then a trajectory of (2) describes the motion of an individual particle in the fluid while the flow of the differential Equation (2) describes the motion of the entire fluid.

3) It can be shown that the basic properties (i)-(iii) of linear flows are also satisfied by nonlinear flows [

4) The following theorem, provides a method of computing derivatives in coordinates.

Given

Thus, if f is a differentiable function, the derivative

An equilibrium

The Hartman-Groβman Theorem [

has the same qualitative structure as the linear system

with

Two autonomous systems of differential equations such as (7) and (8) are said to be topologically equivalent in a neighborhood of the origin or to have the same qualitative structure near the origin if there is a homeomorphism

Consider the linear systems

Let

Then one can easily check that

It then follows that if

The phase plane portraits of the two systems are shown in

Let E be an open subset of

i.e.

In [

Consider the nonlinear system;

The equilibria of the above system is obtained by setting

Solving the above equations we obtain the equilibria as (0, 0) and

To obtain the linearization at the origin, we begin by computing the Jacobian:

Evaluating the Jacobian at the first equilibrium gives

and therefore the linearization of our system at (0, 0) is

Since

and therefore the linearization of our system at

In the simulation which follows we will consider only the nontrivial equilibrium point

a) The given nonlinear system

The solution profiles are depicted in Figures 4(a)-(c).

b) The phase portrait of the linearized system near the origin

The solution profiles are depicted in Figures 6(a)-(c).

The phase portraits of the nonlinear system near

In this section we consider a third order linear equation

which is equivalent to the system

where a, b, c are all positive constants.

We study the asymptotic properties of the above system with the help of MathCAD simulation. The constants a, b, c are chosen such that

The solution matrix for the above system is given in

We now consider the more general nonlinear third order ODE given by

where

We have the following theorem.

Given that

1)

2)

3)

4)

Then every solution

The proof follows that given by Omeike [

Finally, when

We have the following result following immediately from the above theorem.

Given that

1)

2)

3)

Then every solution

1) We note that (1) and (2) are the well known Routh-Hurwitz conditions [

2) For the third order differential equation

In this study, we investigated the stability analysis of certain third order linear and nonlinear ordinary differential equations. We employed the method of phase portrait analysis. We showed, using simulation that the Hartman-Groβman Theorem is verified, for a second order linearized system as an example, approximates the nonlinear system preserving the topological features. In the case of the third order nonlinear system

Okereke, R.N. and Maliki, S.O. (2016) Asymptotic Behaviour of Solutions of Certain Third Order Nonlinear Differential Equations via Phase Portrait Analysis. Applied Mathematics, 7, 2324- 2335. http://dx.doi.org/10.4236/am.2016.718183