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Discrete Dynamical Systems

Dated: 15-06-2025

In this section, we will suppose that \(A\) is diagonalizable1 with \(n\) linearly independent2 eigen vectors3 \(\vec v_1, \ldots, \vec v_n\) and eigen values3 \(\lambda_1, \ldots, \lambda_n\).
Let the eigen vectors3 be arranged in a way such that

\[|\lambda_1| \ge |\lambda_2| \ge \cdots \ge |\lambda_n|\]

Since \(\{\vec v_1, \ldots, \vec v_n\}\) are basis for \(\mathbb R^n\).

\[\vec x_0 = c_1 \vec v_1 + c_2 \vec v_2 + \cdots + c_n \vec v_n\]
\[\vec x_1 = A \vec x_0\]
\[= c_1 A \vec v_1 + c_2 A \vec v_2 + \ldots + c_n A \vec v_n\]
\[= c_1 \lambda_1 \vec v_1 + c_2 \lambda_2 \vec v_2 + \cdots + c_n \lambda_n \vec v_n\]

In general,

\[\vec x_k = c_1 (\lambda_1)^k \vec v_1 + \cdots + c_n (\lambda_n)^k \vec v_n \quad k \ge 0\]

Trajectory of a Dynamical System

Let \(\vec x_0 \in \mathbb R^2\) be an initial value and ran though the transformation4 \(\vec x \to A \vec x\) repeatedly then the graph of \(\{\vec x_1, \vec x_2, \ldots, \vec x_n\}\) is called the trajectory of a dynamical system.

Relationship between eigen values3 and trajectory
  • Origin behaves as attractor if eigen value3 \(< 1\).
  • Origin behaves as repellor if eigen value3 \(> 1\).
  • Origin behaves as saddle point if one eigen value3 \(> 1\) and other eigen value3 is \(< 1\).

Change of Variable

Let \(\{\vec v_1, \ldots, \vec v_n\}\) be the eigen vectors3 serving as basis for \(\mathbb R^n\).
Let \(P = \begin{bmatrix}\vec v_1 & \cdots & \vec v_n\end{bmatrix}\) and \(D\) be the diagonal matrix5 with corresponding eigen values3 on its diagonal.

Given a sequence \(\{\vec x_k\}\) satisfying the relation \(\vec x_{k + 1} = A \vec x_k\), define a new sequence \(\{\vec y_k\}\) such that \(\vec x_k = P \vec y_k\).

\[\vec x_{k + 1} = A \vec x_k\]
\[\because \vec x_k = P \vec y_k\]
\[\vec y_{k + 1} = AP \vec y_k\]
\[\because A = PDP^{-1}\]
\[P\vec y_{k + 1} = (PDP^{-1})P \vec y_k\]
\[P\vec y_{k + 1} = PD(P^{-1}P) \vec y_k\]
\[P\vec y_{k + 1} = PDI \vec y_k\]
\[P\vec y_{k + 1} = PD \vec y_k\]
\[\vec y_{k + 1} = D \vec y_k\]

Denote \(\vec y_k\) as \(\vec y(k)\) and entries in \(\vec y(k)\) as \(\vec y_1(k), \ldots, \vec y_n(k)\)

\[ \begin{bmatrix} \vec y_1(k+1) \\ \vec y_2(k+1) \\ \vdots \\ \vec y_n(k+1) \end{bmatrix} = \begin{bmatrix} \lambda_1 & 0 & \dots & 0 \\ 0 & \lambda_2 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & \dots & 0 & \lambda_n \end{bmatrix} \begin{bmatrix} \vec y_1(k) \\ \vec y_2(k) \\ \vdots \\ \vec y_n(k) \end{bmatrix} \]

Complex Eigenvalues3

Relationship between complex eigen values3 and trajectory

If a matrix6 has 2 complex eigen values3 and absolute value of either eigen value3 is

  • \(> 1\) then origin behaves as a repellor with iterations of \(\vec x_0\) spiraling outwards.
  • \(< 1\) then origin behaves as a attractor with iterations of \(\vec x_0\) spiraling inwards.

References

Read more about notations and symbols.

  1. Read more about diagonalization. ↩

  2. Read more about linear dependency. ↩

  3. Read more about eigen vectors. ↩↩↩↩↩↩↩↩↩↩↩↩↩↩

  4. Read more about linear transformations. ↩

  5. Read more about diagonal matrices. ↩

  6. Read more about matrices. ↩