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2.3 GLOBAL STABILITY

In this section you will investigate

- Conservative Systems
- Reversible Systems
- Lyapunov Functions
- methods of determining the existence of a nonlinear centre

The linearisation theorem tells us that in many cases the solutions of a nonlinear system near an equilibrium point mimic the solutions of the linearisation of the system at that point. However, the result does not apply if the linearisation is non simple or shows a centre. Furthermore if it does apply it tells us only about local behaviour. To make further progress with the analysis of nonlinear systems we need to look at a global analysis. There are certain special systems and methods which we are going to study. These include Conservative Systems, Reversible Systems and Lyapunov Functions.

2.3.1. Conservative Systems

The concept of a conservative dynamical system is important in real world applications. Conservative systems are ones in which a quantity is conserved in the sense that it is constant on any trajectory of the system. Often the conserved quantity has a physical meaning such as the total energy in the case of a system arising in classical mechanics or spin in the case of quantum mechanics. The following definition provides a formal statement of the concept.

If for the system

there exists a function E(x,y) such that its partial derivatives are continuous and E(x,y) satisfies the following conditions

- E(x,y) is constant on a trajectory of the system
i.e.

for all (x,y)

- E(x,y) is non-constant across trajectories.

then E(x,y) is a conserved quantity and the system is conservative.

To verify a function is a conservative quantity for a system.

Here is a worked example to illustrate a method of verifying that a function is a conservative quantity

Worked Example 1

Finding Conserved Quantities using First Integrals

It is difficult to find a conserved quantity E(x,y), One method of doing this is to find a First Integral of the system. First integrals are conserved quantities provided they exist for all (x,y) in the plane.

In the problems that you will meet E(x,y) can be found by finding a First Integral. Here is a worked example to illustrate a method of finding a First Integral.

Worked Example 2

Finding conserved quantities using the total energy

In many dynamical systems the total energy is a conserved quantity. This method is explained and illustrated in worked example 3 in the module coursebook.

Phase portrait of a conservative system.

How can the existence of a conserved quantity for a nonlinear system help us to understand the long term behaviour of that system?

Investigation 1

You are going to compare the global phase portrait of a nonlinear system with the level curves of a conserved quantity for that system. Click on investigation 1 to open the Maple window which contains the material for this investigation

This investigation shows that if we can find a conserved quantity E(x,y) for a nonlinear system then the solution curves of the nonlinear system are the same as the level curves of the conserved quantity.

Stability of Equilibria in a Conservative System

If an equilibrium point also satisfies

and

it is also a stationary point of E(x,y).

To classify the stationary point we need to look at the eigenvalues of the Hessian Matrix ;

If the eigenvalues are

- both positive the stationary point is a local minimum
- both negative the stationary point is a local maximum
- of opposite sign the equilibrium point behaves like a saddle

If an equilibrium point is a local maximum (or minimum) of E(x,y) then the level curves of E(x,y) are closed in the neighbourhood of the equilibrium point. Thus the solutions of the nonlinear system form closed curves surrounding the equilibrium point and the equilibrium point is neutrally stable and a nonlinear centre.

This gives a method of proving the existence of a nonlinear centre.

If the equilibrium point is a saddle for E(x,y) then the equilibrium point is unstable and a nonlinear saddle.

Determination of the existence of a nonlinear centre using a conservative system.

One of the limitations of the linearisation theorem is when the equilibrium point of the linearisation is a centre. It does not follow that the equilibrium point of the nonlinear system is also a centre and further investigation is necessary. We can now do this further analysis for a conservative system.
The method is illustrated in the following example. Click on worked example 4 to open the appropriate Maple window.

Worked example 4.

2.3.2 Reversible Systems.

Many dynamical systems have time-reversed symmetry in the sense that their behaviour is independent of the direction of time. For example a film of a swinging undamped pendulum would appear to be the same whether played forwards or backwards.

Definition: A plane dynamical system

is said to be reversible if the equations are invariant under the change of variables

or

To prove a dynamical system is reversible.

A method of proving a system is reversible is illustrated in example 5 in the handbook.

Phase portrait of a reversible system.

Reversible systems have the property that if {x(t),y(t)}is a solution of the system so is {x(-t),y(-t)}. Thus the phase portrait is symmetrical about either the x-axis or the y-axis.

Stability of equilibria in a reversible system.

Theorem: If a reversible system has an equilibrium point which lies on the axis of symmetry and has a linearisation with a centre then the nonlinear system must have a nonlinear centre.

Thus a reversible system can be useful in proving the existence of a nonlinear centre. The method is illustrated in the example below. Click on worked example 6 to open the relevant Maple window.

Worked example 6

2.3.3 Lyapunov Functions.

Lyapunov Functions are named after the Russian mathematician Alexander Lyapunov.(1857-1918). For a two dimensional system a Lyapunov Function has the following definition. This can be extended to three dimensions.

Definition: A function V(x,y) is a Lyapunov function for a system

provided that along each solution of the system

for every solution (x(t),y(t)) that is not an equilibrium point with strict inequality except for a discrete set of t’s.

To verify a function is a Lyapunov function for a system.

A method of proving a function is a Lyapunov function for a system is illustrated in the example below. Click on worked example 7 to open the Maple window.

Worked example 7

Stability of equilibria using Lyapunov functions.

How can Lyapunov funstions help you to understand the long term behaviour of a nonlinear system?

Investigation2

You are going to draw the phase portrait of a nonlinear system and the level curves of a Lyapunov function for that system. You are going to examine the two diagrams to see if you can see a connection between them. Click on Investigation 2 which opens the Maple window containing the material for this investigation.

As before if the point (p,q) is an isolated equilibrium point of the system and (p,q) is a stationary point of V(x,y) then the level curves of V(x,y) =C are closed curves surrounding (p,q). Since V(x,y) is a decreasing function then as t increases a non equilibrium solution of the system must move across the level curves in a direction in which V(x,y) decreases. This enables the stability of the equilibrium point to be determined and is summarised in the following theorem.

Theorem: If

is a system of equations and V(x,y) is a Lyapunov function for the system that has continuous first and second partial derivatives. Assume that the only way for whichcan remain zero along a solution for any length of time is if the solution is an equilibrium. Then if (p,q) is an isolated equilibrium point and

If the equilibrium point is also a local minimum of V(x,y) then the solution of the system must tend to (p,q) as t approaches infinity. The equilibrium point (p,q) is therefore an attractor and stable.

If the equilibrium point is also a local maximum of V(x,y) then the solution of the system must move away from (p,q) as t approaches infinity and the equilibrium point is a repeller and therefore unstable.

If the equilibrium point is also a saddle point of V(x,y) then the phase portrait of the system also resembles a saddle with the solution curves again crossing the level curves of V.

Note that this method can be used to investigate the stability of equilibrium points for both simple and nonsimple systems of equations.The following example illustratrs the method. Click on worked example 8 to open the Maple window.

Worked example 8