2.5 Asymptotic behaviour

We shall discuss in this section the long term behaviour of solutions of the linear system

$$x' = Ax \, .$$ (2.6)

For any $x_{0} \in R^{n}$, the solution of (2.6) subject to the initial condition x(0) = x₀ is given by

x(t) = e^At x₀.

Clearly, for each fixed t, e^At can be viewed as a mapping from Rⁿ to Rⁿ. The one-parameter family of mappings

$$e^{At} : R^{n} \raro R^{n}, \quad t \in R$$ (2.7)

is called the flow of the linear system (2.6).

It is easy to verify that the flow $\phi_{t} = e^{At}$ satisfies

1.

$\phi_{0} = I$,

2.

$\phi_{t+s} = \phi_{t} \circ \phi_{s}, \quad \forall \, t, s \in R$ (semigroup property).

Definition 2.5.1:     System (2.6) is called

1.

hyperbolic if all the eigenvalues of A have nonzero real parts;

2.

a sink (source) if all the eigenvalues of A have negative (positive) real parts;

3.

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

From Theorem 2.4.4, we see that the following results hold.

Theorem 2.5.1:     The origin is a sink of (2.6) iff $\forall \; x_{0} \in R^{n}$

$$\lim_{t \raro \infty} e^{At} x_{0} = 0 .$$

Theorem 2.5.2:     The origin is a source of (2.6) iff $\forall \; x_{0} \in R^{n}\backslash \{ 0 \}$

$$\lim_{t \raro \infty} \mid e^{At} x_{0} \mid \, = \infty \, .$$

In general, let $\lambda_{j}$, be the eigenvalues of A with multiplicity n_j, $j = 1, \ldots , \gamma$ and ${\dss\sum_{j=1}^{\gamma}} n_{j} = n$. Then corresponding to each $\lambda_{j}$, there are m_j linearly independent eigenvectors and n_j linearly independent generalized eigenvectors with $m_{j}~\leq~n_{j}$. m_j is called the index of $\lambda_{j}$. Clearly, A is diagonalizable iff m_j = n_j for all $j = 1, \ldots , \gamma$. It is easy to see that the real parts of eigenvalues of A determine the asymptotic behaviour of solutions of (2.6). Thus, we give the following definition.

Definition 2.5.2:     Let $\lambda_{j} = \alpha_{j} + i \beta_{j}$, $j=1, \ldots , n$, be the eigenvalues of A repeated according to their multiplicity and w_j = u_j + i v_j, $j=1, \ldots , n$, be the corresponding generalized eigenvectors, where $\beta_{j} =0$ and v_j = 0 if $\lambda_{j}$ is a real eigenvalue. Then we define the stable subspace E^s, centre subspace E^c and unstable subspace E^u of (2.6) by

$$E^{s} = {\rm span} \; \{ u_{j}, v_{j} \, \mid \, \alpha_{j} < 0 \}$$

$$E^{c} = {\rm span} \; \{ u_{j}, v_{j} \, \mid \, \alpha_{j} = 0 \}$$

$$E^{u} = {\rm span} \; \{ u_{j}, v_{j} \, \mid \, \alpha_{j} > 0 \} \, .$$

Clearly, $R^{n} = E^{s} \oplus E^{c} \oplus E^{u}$, the origin is a sink when E^s = Rⁿ, a source when E^u = Rⁿ and a saddle when $E^{c} = \phi$.

From linear algebra, we know that a generalized eigenspace of A, i.e. the span of generalized eigenvectors corresponding to eigenvalues $\lambda$, is A-invariant, i.e. $AE \subset E$. For the linear system (2.6), we have a similar concept.

Definition 2.5.4:     A subspace $E \subset R^{n}$ is said to be an invariant subspace of (2.6) if $e^{At} E \subset E$ for all $t \in R$.

Theorem 2.5.3:     The stable subspace E^s, center subspace E^c and unstable subspace E^u are invariant subspaces of (2.6).

Proof: (omitted)

Thus for all $t \in R$, $e^{At} x_{0} \in E^{s}$ and therefore $e^{At} E^{s} \subset E^{s}$, i.e. E^s is invariant. Similarly, we can show that E^c and E^u are invariant subspaces of (2.6).

Theorem 2.5.4:     If the origin is a saddle point of (2.6), then $R^{n} = E^{s} \oplus E^{u}$ and

1.

$\forall \; x_{0} \in E^{s}$, $e^{At} x_{0} \in E^{s}$, $t \in R$ and ${\dss\lim_{t \raro \infty}} e^{At} x_{0} = 0$.

2.

$\forall \; x_{0} \in E^{u}$, $e^{At} x_{0} \in E^{u}$, $t \in R$ and ${\dss\lim_{t \raro \infty}} \mid e^{At} x_{0} \mid \, = \infty$ if $x_{0} \neq 0$.

Example 2.5.1:     Discuss the asymptotic behaviour of solutions of (2.6) with

$$A = \left[ \begin{array}{rrl} -2 & -1 & 0 \\ 1 & -2 & 0 \\ 0 & 0 & 3 \end{array} \right] \, .$$

Solution:     A has eigenvalues $\lambda_{1} = - 2 + i$, $\lambda_{2} = 3$ and the corresponding eigenvectors

$w_{1} = u_{1} + i v_{1} = \left( \begin{array}{c} 0 \\ 1 \\ 0 \end{array} \right) + i \left( \begin{array}{c} 1 \\ 0 \\ 0 \end{array} \right)$ and $v_{2} = \left( \begin{array}{c} 0 \\ 0 \\ 1 \end{array} \right)$. Then

$$E^{s} = {\rm span} \; \left\{ \left( \begin{array}{c} 0 \\ 1 ... ... \\ 0 \end{array} \right) \right\} = x_{1} x_{2}\mbox{-plane},$$

$$E^{u} = {\rm span} \; \left\{ \left( \begin{array}{c} 0 \\ 0 ... ... \right) \right\} = \, \mbox{the} \quad x_{3}\mbox{-axis, and}$$

the origin is a saddle point. The phase portrait is shown on the right.

Example 2.5.2:     Discuss the asymptotic behaviour of solutions of (2.6) with

$$A = \left[ \begin{array}{lrl} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 2 \end{array} \right] \, .$$

Solution:      $\lambda_{1} =2$, $\lambda_{2} = i$, v₁ = (0,0,1)^T, w₁ = (0,1,0)^T +i (1,0,0)^T.

$$E^{u} = {\rm span} \; \{ (0,0,1)^{T} \} = x_{3}\mbox{-axis}, ... ... \; \{ (0,1,0)^{T}, (1,0,0)^{T} \} = x_{1} x_{2}\mbox{-plane}.$$

The flow e^At is nonhyperbolic. All solutions starting in E^c are bounded and if $x(0) \neq 0$, they are nontrivial periodic solutions

with period $2 \pi$, i.e. $x(t +2 \pi ) = x(t)$ for all $t \in R$. In general, solutions in E^c need not be bounded, e.g.

$$A = \left[ \begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array} \righ... ...uiv c_{1}t + c_{2}. \end{array} \right. \end{array}\right. $}$$

Since the flow e^At is determined by the eigenvalues and the generalized eigenspaces of A, many apparently different linear systems may have flows with identical asymptotic behaviour. Also for any k > 0, e^At and e^kAt have the same phase portrait in the phase space Rⁿ. Thus it is natural to classify all the linear systems by the following definition.

Definition 2.5.5:     Two linear systems x' = A₁ x and x' = A₂x in Rⁿ are said to be linearly equivalent if there exists k > 0 and a nonsingular matrix P such that

A₂ = kP^-1 A₁ P.

By this definition, all linear systems in Rⁿ can be classified into a finite number of equivalent classes. For example, there are 14 equivalent classes in R².

14 equivalent systems in $\mbox{\bm $R^{2}$ }$

$$\mbox{} \hspace*{-.5in}\left. \begin{array}{l} \left. \begin{... ...inear systems} \\ Re (\lambda_{i}) \neq 0 \end{array}\right.$$

$$\left.\begin{array}{llllllll} 10. & \mbox{center} \quad (0) &... ...} \\ Re (\lambda_{1}) Re( \lambda_{2}) = 0 \end{array}\right.$$

Note: The number or symbol in brackets ( ) is the number of nontrivial invariant subspaces.