Matrix Algebra

Dynamical Systems

Dynamical system. Let

be an

-by-

matrix, and let $\mathbf{x}_0$ be an initial vector in $\mathbb{R}^n$ . Then the difference equation

$\displaystyle \mathbf{x}_k = A \mathbf{x}_{k-1}$

is called a dynamical system, which determines recursively how $\mathbf{x}_1, \mathbf{x}_2, \ldots$ evolve as time $k=1,2,\ldots$ changes.

Solution to dynamical system. Suppose that $A = P D P^{-1}$ is diagonalizable with diagonal matrix having diagonal entities $\lambda_1,\ldots,\lambda_n$ . Then we have $A^k = P D^k P^{-1}$ , and the -th power is simply expressed by

$\displaystyle D^k = \begin{bmatrix} \lambda_1^k & 0 & \cdots & 0 \\ 0 & \la... ...\cdots & 0 \\ \hdotsfor{4} \\ 0 & 0 & \cdots & \lambda_n^k \end{bmatrix}$

Furthermore, the vector $\mathbf{x}_k$ satisfying the difference equation is obtained in the form of

$\displaystyle \mathbf{x}_{k} = A^k \mathbf{x}_{0} = P D^k P^{-1} \mathbf{x}_{0} = c_1 \lambda_1^k \mathbf{v}_{1} + \cdots + c_n \lambda_n^k \mathbf{v}_{n},$ $P^{-1} \mathbf{x}_{0} = \begin{bmatrix}c_1 [-0.05in] \vdots [-0.05in] c_n \end{bmatrix}$ . $\displaystyle$

Description of dynamical systems. Let be a $2 \times 2$ matrix. By choosing an initial vector $\mathbf{x}_0 = \begin{bmatrix}x_0 \\ y_0 \end{bmatrix}$ , we can obtain the trajectory $\mathbf{x}_1,\ldots,\mathbf{x}_t$ of dynamical system recursively for $k = 1,\ldots,t$ . Then the trajectory are plotted on the -plane to see graphically how the system evolves, called a phase portrait. And it can be repeated with different choices of initial vector $\mathbf{x}_0$ to get a geometric description of the system.

EXAMPLE 1. The phase portrait below demonstrates that a dynamical system has an attractor at the origin when both eigenvalues are less than one in magnitude.

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EXAMPLE 2. A dynamical system will exhibit a repellor at the origin of phase portrait when both eigenvalues are larger than one in magnitude.

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EXAMPLE 3. A dynamical system will exhibit a saddle point at the origin when one eigenvalue is less than one and the other is larger than one in magnitude.

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EXAMPLE 4. Observe that the eigenvalues for the matrix in Examples 3 and 4 are exactly the same, and that the dynamical system in Example 4 has a saddle point at the origin. This indicates that the behavior of dynamical systems (attractor, repellor, or saddle point) is determined by the eigenvalues of .

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EXAMPLE 5. A rotation is introduced in the phase portrait when the eigenvalues of are complex numbers.

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