3.7 Limit sets and long-term behaviour

We will introduce, in this section, an important concept called limit set, which is invariant. The invariant property of a limit set will enable us to weaken the conditions on asymptotic stability given in Theorem 3.5.1.

Consider the nonlinear system

 

x' = f(x) (3.29)

we assume that the system (3.29) determines a flow $\psi_{t} : \Omega \raro \Omega$ $t \in R$. For $x_{0} \in \Omega$, we denote by $x(t) = \psi_{t} (x_{0})$ the solution of (3.29) satisfying x(0) = x₀.

Definition 3.7.1:     Let $x_{0} \in \Omega$ and $x(t) = \psi_{t} (x_{0})$ be the solution of (3.29). Then a point $p \in \Omega$ is called an $\omega$-limit ($\alpha$-limit) point x₀ if there exists a sequence $\{ t_{n} \}$ $(\{ t_{-n}\} )$, $t_{n} \raro \infty$ as $n \raro \infty$ $(t_{-n} \raro - \infty$ as $n \raro \infty$) such that

$$\lim_{n \raro \infty} x(t_{n}) = p \quad (\lim_{n \raro \infty} x(t_{-n}) = p) \, .$$

The set of all $\omega$-limit ($\alpha$-limit) points of x₀ is called the $\omega$-limit ($\alpha$-limit) set of x₀, denoted by $\omega (x_{0})$ $(\alpha (x_{0}))$.

By Definition 3.7.1, it is clear that if $\ol{x}$ is an asymptotically stable equilibrium, it is the $\omega$-limit set of every point in its basin of attraction. Any equilibrium is its own $\omega$-limit set and $\alpha$-limit set. A closed orbit is the $\omega$-limit and $\alpha$-limit set of every point on it.

Example 3.7.1:     Consider the nonlinear system in R²

$$\left\{ \begin{array}{l} x_{1}' = x_{1} (1-x_{1}), \\ x_{2}' = - x_{2}, \end{array} \right.$$

which has equilibrium points (0,0) and (1,0). It is easy to see that for x₀ = (x₁₀, x₂₀)

$$\omega (x_{0}) = \left\{ \begin{array}{cl } \{ (1,0) \} , & \... ...x_{10} = 0, \\ \phi , & \mbox{otherwise}. \end{array} \right.$$

Example 3.7.1 Simulation

Example 3.7.2:     Consider the magneto-elastic beam. Let q denote the displacement of the tip of the beam from the line of symmetry. Set, for simplicity, the mass of the beam m=1. Then the force (elastic +magnetic) is

$$F(q) = \beta^{2} q - q^{3} \, .$$

Thus $V(q) = - \frac{1}{2} \beta^{2} q^{2} + \frac{1}{4} q^{4}$, set x₁ = q x₂ = q'. Then $H (x) = \frac{1}{2} x_{2}^{2} - \frac{1}{2} \beta^{2} x_{1}^{2} + \frac{1}{4} x_{1}^{4}$ and

$$\left\{ \begin{array}{l} x_{1}' = x_{2}, \\ x_{2}' = \beta^{2} x_{1} - x_{1}^{3}. \end{array} \right.$$

Let x₀ = (x_i, 0), i=1,2,3, as shown. Then we see

$$\omega (x_{0}) = \alpha (x_{0}) = \left\{ \begin{array}{ll} \... ... \\ \gamma_{3}, & x_{0} = (k_{3}, 0) \, . \end{array} \right.$$

     

Example 3.7.2 Simulation

It should be noted that an $\omega$-limit set $\omega (x_{0})$ with respect to a flow $\{ \psi_{t} \}$ is an $\alpha$-limit set $\alpha (x_{0})$ with respect to the flow $\{ \psi_{-t} \}$. Thus we only need to discuss the properties of $\omega$-limit set. Clearly, the long-term behaviour of the physical system whose evolution is governed by (3.29) is described by the $\omega$-limit set $\omega (x_{0})$ of the given initial state $x_{0} \in \Omega$. In order to determine the $\omega$-limit set, it is helpful to know a positively invariant set since orbits entering a positively invariant set never leave it, i.e. if E is invariant, then $\forall \; x_{0} \in E$, $\omega (x_{0}) \subset E$.

Theorem 3.7.1:     Let $f \in C^{1} (\Omega )$ and $x_{0} \in \Omega$. Then

(1)

$\omega (x_{0}) \neq \phi$ if $\psi_{t} (x_{0})$ is bounded for all $t \geq 0$;

(2)

$\omega (x_{0}) \neq \phi$ implies that $\omega (x_{0})$ is a whole orbit or the union of more than one whole orbit.

Proof: (omitted)

    Here is a phase portrait of a nonlinear system to illustrate the second possibility in the theorem.

$$\omega (x_{0}) = \gamma_{1} \cup \gamma_{2} \cup \gamma_{3} \... ...mbda x + \frac{1}{2} (x^{2} - y^{2}) - cy \end{array} \right.$$

$$\mbox{} \hspace*{2in} \left. \begin{array}{l} H = - \frac{\l... ...2} \left( xy^{2} - \frac{x^{3}}{3} \right) \end{array} \right.$$

which is a union of six orbits.

Corollary 3.7.1:     If $\Omega \subset R^{n}$ is a bounded and positively invariant set of (3.29), then $\forall \; x_{0} \in\Omega$, $\omega (x_{0}) \neq \phi$ and $\omega (x_{0})$ is invariant.

In order to determine the $\omega$-limit sets, we again resort to Lyapunov functions. The idea is that if $\dot{V} (x) = \nabla V \cdot f \leq 0$ for all $x \in \Omega$ then V(x) will keep decreasing along any solution $\psi_{t} (x_{0})$ staying in $\Omega$ for all $t \geq 0$ until the solution approaches its $\omega$-limit set $\omega (x_{0})$. Thus we expect $\omega (x_{0}) \subset \{ x \in \Omega , \, \dot{V} (x) = 0 \}$, which is a strong restriction on the possible $\omega$-limit sets. Based on this observation and the invariance property of the limit set, Barbashin and Krasovski generalized, in 1952, Lyapunov's theorem (Th. 3.5.1) on asymptotic stability to the case when the derivative of a Lyapunov function along solutions of (3.29) is only semi-negative definite for periodic differential equations. Inspired by their results, LaSalle considered autonomous systems and made a number of contributions. The general result is now known as LaSalle's invariance principle.

Theorem 3.7.2:     Let $V : D \raro R$, $D \subset R^{n}$, be continuously differentiable and bounded from below . Assume that

(i)

$\dot{V}(x) = \nabla V(x) \cdot f(x) \leq 0, \quad x \in \Omega , \quad \ol{\Omega} \subset D$;

(ii)

$x (t, x_{0}) = \psi_{t} (x_{0})$ is a solution of (3.29) such that

$$x(t, x_{0}) \in \Omega \quad \mbox{for all}\quad t \in [0, \infty ) \, .$$

Then for some real number c, $\omega (x_{0}) \subseteq E \cap V^{-1} (c)$, where

$$E = \{ x \in \ol{\Omega}; \; \dot{V} (x) = 0 \} \quad \mbox{and} \quad V^{-1} (c) = \{ x \in \ol{\Omega}; \; V(x) = c \} \, .$$

Proof: (omitted)

Since $\omega (x_{0})$ contains only whole orbits of (3.29), the next result follows easily from Theorem 3.7.2 and Corollary 3.7.1.

Corollary 3.7.2:     Assume that

(i)

$\Omega \subset R^{n}$ is a bounded and positively invariant set of (3.29);

(ii)

$V : R^{n} \raro R$ is continuously differentiable, bounded from below and

$$\dot{V} (x) \leq 0, \quad x \in \Omega ;$$

(iii)

the set $E = \{ x \in \ol{\Omega} ; \quad \dot{V} (x) = 0 \}$ does not contain any whole orbits except the equilibrium point $\ol{x}$ of (3.29).

Then $\forall \; x_{0} \in\Omega$, $\psi_{t} (x_{0}) \raro \ol{x}$ as $t \raro \infty$.

In view of Theorem 3.5.1 and Corollary 3.7.2, we obtain the following result which gives asymptotic stability under weaker conditions.

Theorem 3.7.3:    Let $\ol{x}$ be an equilibrium point of (3.29), $\Omega \subset R^{n}$ be an open set containing $\ol{x}$ and $V : R^{n} \raro R$ be continuously differentiable such that

(a)

$V (\ol{x}) = 0$ and V(x) > 0 if $x \neq \ol{x}$;

(b)

$\dot{V} (x) \leq 0$, $x \in \Omega$;

(c)

the set $E = \{ x \in \ol{\Omega} ; \quad \dot{V} (x) = 0 \}$ does not contain any whole orbits except $\ol{x}$.

Then $\ol{x}$ is asymptotically stable.

Example 3.7.3:     Consider the nonlinear system

$$\left\{ \begin{array}{l} x' = y \\ y' = - x^{2} y -x \, , \end{array} \right.$$

which has a unique equilibrium point (0,0). It is easy to check that (0,0) is nonhyberbolic and thus linearization fails to provide even local information about the solutions of the nonlinear system. However, if we choose $V (x,y) = \frac{1}{2} x^{2} + \frac{1}{2} y^{2}$, then

$$\dot{V} (x,y) = - x^{2} y^{2} \leq 0 \, .$$

Clearly, for any constant M > 0, the set

$$\ol{\Omega} = \{ (x,y) \in R^{2}; \quad V(x,y) \leq M \}$$

is positively invariant, bounded and contains (0,0).

$$E = \{ (x,y) \in \ol{\Omega} ; \quad \dot{V} (x,y) = 0 \} = \... ... \ol{\Omega} ; \quad x = 0 \quad \mbox{or} \quad y = 0 \} \, .$$

Since (0,0) is the only whole orbit contained in E, it follows from Corollary 3.7.2 that all solutions starting in $\Omega$ converges to (0,0) as $t \raro \infty$. It is easy to see, in view of the function V(x,y), that (0,0) is stable. Thus we conclude that (0,0) is asymptotically stable.

It should be noted that since the constant M is arbitrary, $\psi_{t} (x_{0}, y_{0}) \raro (0,0)$ as $t \raro \infty$ for any $(x_{0}, y_{0}) \in R^{2}$, i.e. the domain of the attraction of (0,0) is R². This is often called that (0,0) is globally asymptotically stable.

Example 3.7.3 Simulation

Example 3.7.4:    Consider the predator prey system

$$\left\{\begin{array}{l} x' = x(3-x-y), \\ y' = y(-1 +x-y) \, . \end{array} \right.$$

This system has three equilibria (0,0), (3,0) and (2,1). Linearization shows that (0,0) and (3,0) are saddles and (2,1) is a sink.

For x > 0 and y > 0, let $V(x,y) = x - 2 \ln x + y - \ln y$. Then V(x,y) >0 for x,y > 0 and

$$\begin{eqnarray*}\dot{V} (x,y) & = & x' - 2 \frac{x'}{x} + y' - \frac{y'}{y} = x... ...-4x-2y+x^{2} +y^{2}) \\ \\ & = & -(x-2)^{2} -(y-1)^{2} \leq 0 \end{eqnarray*}$$

$E = \{ (x,y)$; $\dot{V} (x,y) = 0 \} = \{ (2,1) \}$. Since V(x,y) is monotonely increasing in x (y) for y > 0 (x >0), it follows that all solutions of the nonlinear system are bounded. Choose $\Omega = \{ (x,y) \in R^{2}$, $x > 0, \; y > 0 \}$. Then by Theorem 3.7.2, the equilibrium point (2,1) is asymptotically stable and its domain of attraction is $\Omega$.

Example 3.7.4 Simulation

Example 3.7.5:     Consider the damped magneto-elastic beam

$$\left\{ \begin{array}{ll} x_{1}' = x_{2}, &\\ x_{2}' = \beta... ...} - x_{1}^{3} - cx_{2}, & \beta, c > 0 \, .\end{array} \right.$$

The total energy function $H (x_{1}, x_{2}) = \frac{1}{2} x_{2}^{2} - \frac{1}{2} \beta^{2} x_{1}^{2} + \frac{1}{4} x_{1}^{4}$

$$\dot{H} (x_{1}, x_{2}) = - cx_{2}^{2} \, .$$

Thus $E = \{ (x_{1}, x_{2}) \in R^{2}$; $\dot{H} (x_{1}, x_{2}) = 0 \} = \{ (x_{1}, x_{2}) \in R^{2}$; $x_{2} = 0 \}$, which contains three whole orbits, namely three equilibria (0,0), $(\pm \beta, 0)$. Thus all solutions converge to the set $\{ (0,0), \, (\pm \beta , 0) \}$ as $t \raro \infty$. Linearization shows that (0,0) is a saddle point. Let $V(x_{1}, x_{2}) = H (x_{1}, x_{2}) + \frac{1}{4} \beta^{4}$. Then $V(x_{1}, x_{2}) = \frac{1}{2} x_{2}^{2} + \frac{1}{4} (x_{1}^{~2} - \beta )^{2}$, and

$$\dot{V} (x_{1}, x_{2}) = - c x_{2}^{2} \, .$$

Let

$$\Omega_{1} = \left\{ (x_{1}, x_{2}) \in R^{2} ; \, x_{1} < 0 \; \mbox{and}\; V(x_{1}, x_{2}) < \frac{\beta^{4}}{4} \right\} ,$$

$$\Omega_{2} = \left\{ (x_{1}, x_{2}) \in R^{2} ; \, x_{1} > 0 ... ...ox{and} \; V(x_{1}, x_{2}) < \frac{\beta^{4}}{4} \right\} \, .$$

Then V (x₁, x₂) is positive definite with respect to $(- \beta , 0)$ in $\Omega_{1}$ and positive definite with respect to $(\beta , 0)$ in $\Omega_{2}$. $\Omega_{1}$ and $\Omega_{2}$ are both positively invariant. Thus the stable manifold equilibrium points $(- \beta , 0)$ and $(\beta , 0)$ are asymptotically stable with domains of attraction $\Omega_{1}$ and $\Omega_{2}$ respectively.

Example 3.7.5 Simulation

Remark:     In applying the theorem, one must keep in mind that the $\omega$-limit set $\omega (x_{0})$ is a whole orbit or the union of whole orbits. Thus one has to find the whole orbits in the set $\{ x \in \Omega \mid \dot{V} (x) = 0 \}$.

Example 3.7.6:     Consider the damped pendulum

$$\left\{ \begin{array}{ll} x_{1}' = x_{2} & \\ x_{2}' = - \al... ...> 0 \quad \mbox{is the damping parameter.} \end{array} \right.$$

Let $V (x) = \frac{1}{2} x_{2}^{2} + (1-\cos x_{1})$. Then $V (x) \geq 0$ and

$$\dot{V} (x) = - \alpha x_{2}^{2} \leq 0 \, .$$

On the state space $S^{1} \times R$, $S^{1} = [ - \pi , \pi ]$, the level sets of V are shown below.

Clearly $S_{K} = \{ x \mid V(x) \leq k \}$ is positive invariant. Thus $\forall \; x_{0} \in S_{K}$, by Corollary 3.7.2

$$\omega (x_{0}) \subseteq \{ x \in S_{K}, \dot{V} (x) = 0 \} = \{ x \in S_{K}; x_{2} = 0 \} \, .$$

The only possible whole orbits with x₂ = 0 are the equilibrium points (0,0) and $(\pm \pi , 0)$. It follows that $\omega (x_{0}) \subseteq \{ (0,0), (\pm \pi , 0) \}$. But by linearization, $(\pm \pi , 0)$ are saddle points. It follows from the stable and unstable Manifold Theorem that there are precisely two orbits which tend to $(\pm \pi , 0)$ as $t \raro \infty$. Thus for all initial states not on these orbits, e.g. V(x₀) < 2, then

$$\omega (x_{0}) = \{ (0,0) \} \, .$$

This proves that the equilibrium point (0,0) is asymptotically stable.

Example 3.7.6 Simulation

To conclude this section, it should be noted that although the invariance property of the limit sets enables us to weaken Lyapunov's theorem on asymptotic stability, the structure of limit sets can be very complicated, and far from being understood, e.g. the ``strange attractor'' for the Lorenz system, see Perko, P_180-181. In fact, it is still a very active area of current research especially for higher $(n \geq 3)$ dimensional systems. For two dimensional systems, the limit sets are fairly simple and will be discussed in depth in the next chapter.