3.1 Nonlinear sinks and sources

In this section, we consider the nonlinear systems of ordinary differential equations

 

x' = f(x) (3.1)

where $x \in R^{n}$, $f : \Omega \raro R^{n}$ and $\Omega$ is an open subset of Rⁿ. As mentioned earlier, it is not possible, in general, to solve the nonlinear system (3.1). However, we will show in the next section, that under certain assumptions of f, the nonlinear system (3.1) has a unique solution existing on R through each point $x_{0} \in \Omega$. To give you a quick glance of the applications of the linear theory developed in the last chapter to nonlinear systems, we assume, for simplicity, that $\Omega = R^{n}$, f is continuously differentiable and for any $x_{0} \in R^{n}$, there exists a unique solution x(t) of (3.1) existing on R satisfying x(0) = x₀. We often denote x(t) by $\phi_{t}(x_{0})$.

Theorem 3.1.1:     Let $\ol{x}$ be an equilibrium point of (3.1) and suppose that all eigenvalues of the matrix $A = Df (\ol{x})$ have negative real parts. Then there exists a neighbourhood U about $\ol{x}$ such that for some constant M, k > 0

$$\mid \phi_{t} (x) - \ol{x} \mid \, \leq M e^{-kt} \mid x - \o... ...\mid \, , \quad \forall \quad x \in U \, , \quad t \geq 0 \, .$$

We need the following Lemma whose proof can be found in the book by Hirsch/Smale, P. 145-9.

Lemma 3.1.1:     Let A be an $n \times n$ matrix and $\alpha < Re (\lambda ) < \beta$ for all eigenvalues $\lambda$ of A. Let $B = \{ v_{1}, \ldots v_{n} \}$ be a basis for Rⁿ. Then for any $x, y \in R^{n}$, $x = {\dss\sum_{i=1}^{n}} a_{i} v_{i}$ and $y = {\dss\sum_{i=1}^{n}} b_{i} v_{i}$.

(i)

$\mid x \mid_{B} = \sqrt{{\dss\sum_{i=1}^{n}} a_{i}^{2}}$ defines a norm in Rⁿ and $\langle x, y \rangle_{B} = {\dss\sum_{i=1}^{n}} a_{i} b_{i}$ defines an inner product in Rⁿ.

(ii)

there exists a basis B in Rⁿ such that for any $x \in R^{n}$

$$\alpha \mid x \mid_{B}^{2} \, \leq \langle x, Ax \rangle_{B} \leq \beta \mid x \mid_{B}^{2} \, .$$

Proof of Theorem 3.1.1: (omitted)

Example 3.1.1: Consider the nonlinear oscillator (hard spring)

$$\left\{ \begin{array}{l} x'_{1} = x_{2} \\ x'_{2} = - x_{1} ... ...^{3} - \alpha x_{2}, \quad \alpha > 0 \, . \end{array} \right.$$ (3.3)

It is easy to verify that (0,0) is the only equilibrium point, at which the linearization of (3.3) is

$$y' = Df (0,0) y \,, \quad \mbox{where} \quad Df (0,0) = \left... ...gin{array}{rr} 0 & 1 \\ -1 & - \alpha \end{array} \right] \, .$$

It is easy to compute that Df(0,0) has eigenvalues $\lambda = {\dss\frac{- \alpha \pm \sqrt{\alpha^{2} - 4}}{2}}$ which have negative real parts. Thus by Theorem 3.1.1, we conclude that there exists a neighbourhood U , of the origin such that all solutions (x₁ (t), x₂(t)) of (3.3) starting in U satisfying

$$\sqrt{x_{1}^{2} (t) + x_{2}^{2} (t)} \leq Me^{-kt} \sqrt{x_{1}^{2} (0) + x_{2}^{2} (0)} \, , \quad t \geq 0$$

for some M > 0 and $0 < k < \alpha$.

Naturally, we are interested in finding a neighbourhood U of the origin in which all solutions of (3.3) tend to (0,0) as $t \raro \infty$. By Theorem 2.6.2, we see that there exists a positive definite matrix Q such that, letting A = Df (0,0),

$$A^{T} Q + QA = - I \, .$$

Solving the above Lyapunov equation, we obtain

$$Q = \left[ \begin{array}{ll} \left( \frac{1}{\alpha} + \frac{... ...2} \\ \frac{1}{2} & \frac{1}{\alpha} \end{array} \right] \, .$$

Let

$$V(x_{1}, x_{2}) = x^{T} Q x = (x_{1} x_{2}) \left( \begin{arr... ...ght) x_{1}^{2} + x_{1} x_{2} + \frac{1}{\alpha} x_{2}^{2} \, .$$

The derivative of V along solutions of (3.3) is

$$\begin{eqnarray*}\frac{dV}{dt} & = & 2 \left( \frac{1}{\alpha} +\frac{\alpha}{2}... ...\frac{1}{\alpha} x_{1}^{2} \right) (x_{1}^{2} + x_{2}^{2} ) \, . \end{eqnarray*}$$

Thus for $\gamma < \sqrt{\alpha}$, $\sqrt{x_{1}^{2} + x_{2}^{2}} < \gamma$ implies

$$\frac{dV}{dt} \leq - \left( 1 - \frac{\gamma^{2}}{\alpha} \ri... ..., \quad k = 1 - \frac{\gamma^{2}}{\alpha} \, ,\quad t \geq 0$$

for any solution (x₁(t), x₂(t)) of (3.3).

Thus $\forall \quad (x_{1}, x_{2}) \in U = \{ (x_{1}, x_{2}) \in R^{2}, \quad x_{1}^{2} + x_{2}^{2} < \gamma^{2} \}$,

$$\phi_{t} (x_{1}, x_{2}) \raro 0 \quad \mbox{as} \quad t \raro \infty \, .$$

This example motivates the following result.

Theorem 3.1.2:     Let $\ol{x}$ be an equilibrium point of the system (3.1) and suppose that all eigenvalues of the matrix $A = Df (\ol{x})$ have negative real parts. Then there exists a quadratic function V(x) and a neighbourhood U about $\ol{x}$ such that $V (x) \geq 0$, V(x) = 0 iff $x = \ol{x}$ and

$$\frac{dV(x(t))}{dt} = \nabla V (x) f(x) \leq - k \mid x (t) -... ...d \forall \quad x \in U \quad \mbox{for some} \quad k > 0 \, .$$

Proof: (omitted)

Theorem 3.1.3:     Let $\ol{x}$ be an equilibrium point of the system (3.1) and suppose that all eigenvalues of the matrix $A = Df (\ol{x})$ have positive real parts. Then

(i)

there exists a neighbourhood U about $\ol{x}$ and constants $L, \alpha > 0$ such that

$$\mid \phi_{t} (x) - \ol{x} \mid \, \geq L e^{\alpha t} \mid x... ...x} \mid , \quad \forall \quad x \in U \, , \quad t \geq 0 \, ,$$

(ii)

there exists a neighbourhood U about $\ol{x}$, a quadratic function V(x), $V (x) \geq 0$ V(x) = 0 iff $x = \ol{x}$ and a constant $\sigma > 0$ such that for all solutions x(t) of (3.1) in U

$$\frac{dV(x(t))}{dt} \geq \sigma \mid x (t) - \ol{x} \mid^{2} ... ...mbox{as long as} \quad x(t) \quad \mbox{stays in} \quad U \, .$$

Theorems 3.1.1 - 3.1.3 tell us that solutions of the nonlinear system (3.1) near the equilibrium point $\ol{x}$ behave exactly as those of the corresponding linear system at $\ol{x}$. Thus we give the following definition.

Definition 3.1.1:     Let $\ol{x}$ be an equilibrium point of (3.1). Then $\ol{x}$ is called

1.

hyperbolic if no eigenvalues of the matrix $Df(\ol{x})$ have zero real parts;

2.

a nonlinear sink if all eigenvalues of the matrix $Df(\ol{x})$ have negative real parts;

3.

a nonlinear source if all eigenvalues of the matrix $Df(\ol{x})$ have positive real parts;

4.

a saddle if it is hyperbolic but neither a sink nor a source.

We will discuss solutions of (3.1) near a saddle point in section 3.4.