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# Stability of equilibrium

Module by: Aurèle Balavoine. E-mail the author

Summary: This modules defines the notions of equilibrium or fixed point of an Ordinary Differential Equation along with the notions of stability, uniform stability, asymptotic stability and global asymptotic stability.

Read more about ordinary differential equations, initial value problem and Lyapunov direct method.

In the study of differential equations, it is important to determine the stability of equilibrium points. For real systems, such a study allows one to determine the effect of a small perturbation around the equilibrium. If the system is stable, then the trajectories will stay within a small neighborhood of the equilibrium. On the other hand, perturbations around an unstable equilibrium generate trajectories that move away from the equilibrium.

## Equilibrium points

Let tt be a positive scalar and xx in RNRN a vector of dimension NN. In this section, we are interested in an autonomous ordinary differential equation of the form:

x ˙ ( t ) = f ( x ( t ) ) . x ˙ ( t ) = f ( x ( t ) ) .
(1)

The function f:DRNf:DRN is continuous function on DD, a subset of RNRN, and let x˙=dx/dtx˙=dx/dt.

Definition 1.1 An equilibrium or fixed point of Equation 1 is a constant solution x(t)=x*x(t)=x* that satisfies:

x ˙ ( t ) = 0 on I . x ˙ ( t ) = 0 on I .
(2)

## Stability of an equilibrium

There exist several notions of stability that describe the evolution of solutions near an equilibrium both locally and globally. The notion of Lyapunov stability is local and guarantees that if a solution starts close to an equilibrium, it will remain nearby.

Definition 2.1 An equilibrium x*x* of Equation 1 is Lyapunov stable if for any ϵ>0ϵ>0 and any t00t00, there exists δ>0δ>0 such that for any point (t0,x0)D(t0,x0)D such that x0-x*<δx0-x*<δ, and for all the solutions xx to Equation 1 with initial condition (t0,x0)(t0,x0), the following holds

x ( t ) - x * < ϵ , t t 0 . x ( t ) - x * < ϵ , t t 0 .
(3)

This property is illustrated on Figure 1.

A stronger notion of stability is the following.

Definition 2.2 An equilibrium x*x* of Equation 1 is uniformly stable if for any ϵ>0ϵ>0, there exists δ>0δ>0 such that for any point (t0,x0)D(t0,x0)D such that x0-x*<δx0-x*<δ, and for all the solutions xx to Equation 1 with initial condition (t0,x0)(t0,x0), the following holds

x ( t ) - x * < ϵ , t t 0 . x ( t ) - x * < ϵ , t t 0 .
(4)

Notice that, for an equilibrium to be uniformly stable, the parameter δδ must be chosen independently of t0t0.

On the other hand, the notion of instability is simply defined as follows.

Definition 2.3 An equilibrium x*x* of Equation 1 is unstable if it is not stable.

## Example

Sometimes, the phase portrait is sufficient to determine the stability of an equilibrium. Take for example the differential equation:

x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = sin ( x 1 ( t ) ) . x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = sin ( x 1 ( t ) ) .
(5)

The phase portrait is plotted on Figure 2. Since the trajectories are tangent to and in the direction of the vector field f(x1,x2)=(x2,sin(x1))f(x1,x2)=(x2,sin(x1)), it can be seen from the phase portrait that the point x*=(0,0)x*=(0,0) is unstable (trajectories close to it move away), while the point x*=(0,π)x*=(0,π) is stable (trajectories close to it move in a circle of fixed radius around it). A few rajectories for different initial conditions are plotted in red to illustrate this fact.

## Asymptotic stability

Another notion of stability, called asymptotic stability, describes how solutions that start near an equilibrium converge to the equilibrium.

Definition 4.1 An equilibrium x*x* of Equation 1 is asymptotically stable or attracting if it is Lyapunov stable and if for any t00t00, there exists δ>0δ>0 such that for any point (t0,x0)D(t0,x0)D such that x0-x*<δx0-x*<δ, and for all the solutions xx to Equation 1 with initial condition (t0,x0)(t0,x0), the following holds

lim t x ( t ) - x * = 0 . lim t x ( t ) - x * = 0 .
(6)

This is a local notion of stability but it can be extended to a global property known as global asymptotic convergence, in which case solutions converge to an equilibrium regardless of the initial condition.

Definition 4.2 An equilibrium x*x* of Equation 1 is globally asymptotically stable if it is Lyapunov stable and if for any point (t0,x0)D(t0,x0)D, and for all the solutions xx to Equation 1 with initial condition (t0,x0)(t0,x0), the following holds

lim t x ( t ) - x * = 0 . lim t x ( t ) - x * = 0 .
(7)

## Example

In the previous example, the equilibrium x*=πx*=π was stable. However, it is not asymptotically stable. Indeed, all the trajectories that start close to x*=(0,π)x*=(0,π) stay at a fixed distance from the equilibrium and they get close to x*x* as time increases. We can slightly modify the differential equation in order to create an asymptotically stable equilibrium. Consider the new differential equation:

x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = sin ( x 1 ( t ) ) - x 2 ( t ) . x ˙ 1 ( t ) = x 2 ( t ) x ˙ 2 ( t ) = sin ( x 1 ( t ) ) - x 2 ( t ) .
(8)

The associated phase portrait is plotted on Figure 3, along with some example trajectories. This time, it can be seen that the point x*=(0,π)x*=(0,π) is asymptotically stable since trajectories that start close by end up converging towards the equilibrium.

## Attractors

Sometimes, solutions to differential equations like Equation 1 converge to something more complex than a single point, such as a circle, an ellipse or some other trajectory. In those cases, we talk about an ωω-limit set or an attractor for the dynamical system, that is to say the set of points that the dynamical system keeps visiting as time evolves.

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