Discrete dynamical systems - Eigenvalue and Eigenvectors

Discrete dynamical systems

Lessons

Notes:

Assume that $A$ is diagonalizable, with $n$ linearly independent eigenvectors $v_1, v_2, \cdots , v_n$ , and corresponding eigenvalues $\lambda _1, \lambda _2, \cdots \lambda _n$ . Then we can write an initial vector $x_0$ to be:
$x_0=c_1 v_1+c_2 v_2+ \cdots +c_n v_n$
Let's say we want to transform $x_0$ with matrix $A$ . Let's call the transformed vector to be $x_1$ . Then, $x_1=Ax_0=c_1 Av_1+c_2 Av_2+\cdots+c_n Av_n$
$=c_1 \lambda_1 v_1+c_2 \lambda_2 v_2+\cdots+c_n \lambda_n v_n$
Let's say we want to keep transforming it with matrix $A\; k$ times. Then we can generalize this to be:
$x_k=c_1 (\lambda_1 )^k v_1+c_2 (\lambda_2 )^k v_2+\cdots+c_n (\lambda_n )^k v_n$
This is useful because we get to know the behaviour of this equation when $k$ → $\infty$ .

Intro Lesson

Discrete Dynamical Systems Overview:
4.
Predator and Prey Model
Let the eagle and rabbit population at time $k$ be denoted as , where $k$ is the time in years, $E_k$ is the number of eagles at time $k$ , and $R_k$ is the number of rabbits at time $k$ (all measured in thousands). Suppose there are two equations describing the relationship between these two species:

$E_{k+1}=(.4) E_k+(.5)R_k$
$R_{k+1}=(-.207) E_k+(1.2) R_k$

Discrete dynamical systems

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Lessons

Notes:

Intro Lesson

Discrete Dynamical Systems Overview:

a)

The Differential Equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​ • Linear Transformation • Multiplying with A k times • Generalized formula • Why is this formula useful?

b)

Doing an Example • Calculate xkx_kxk​ • Long-term behaviour (k(k (k→∞) \infty)∞)

c)

Application: Predator and Prey Model • Analyzing the predator and prey equations • Converting to the equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​ • Calculating the general solution • Long term behaviour (k(k (k→∞) \infty)∞)

1.

Finding the General Solution Let AAA be a 3×33 \times 33×3 matrix with eigenvalues 3,2,3,2,3,2, and 12\frac{1}{2}21​, and corresponding eigenvectors and . if , find the general solution of the equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​.

2.

Analyzing the Long Term Behaviour Explain the long term behaviour (k(k (k→∞) \infty)∞) of the equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​, where And where c1c_1c1​ > 000 and c2c_2c2​ > 000.

3.

Explain the long term behaviour (k(k (k→∞) \infty)∞) of the equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​, where And where c1c_1c1​ > 000 and c2c_2c2​ > 000

4.

a)

What happens to the rabbits if there are no eagles? What happens to the eagles if there are no rabbits?

b)

Find the matrix AAA for the equation xk+1=Axkx_{k+1}=Ax_kxk+1​=Axk​.

c)

Suppose the eigenvalues of AAA is λ1=1.0377\lambda_1=1.0377λ1​=1.0377 and λ2=0.562303\lambda_2=0.562303λ2​=0.562303 and the corresponding eigenvectors are and (assume a,b,c,da,b,c,da,b,c,d > 000). Find the general solution.

d)

Assuming that c1,c2c_1,\;c_2c1​,c2​ > 000, explain the population of rabbits and eagles as kk k→∞ \infty∞.

Discrete dynamical systems

Don't just watch, practice makes perfect.

We have over 70 practice questions in Linear Algebra for you to master.

or

The Differential Equation $x_{k+1}=Ax_k$
• Linear Transformation
• Multiplying with A k times
• Generalized formula
• Why is this formula useful?

Doing an Example
• Calculate $x_k$
• Long-term behaviour $(k$ → $\infty)$

Application: Predator and Prey Model
• Analyzing the predator and prey equations
• Converting to the equation $x_{k+1}=Ax_k$
• Calculating the general solution
• Long term behaviour $(k$ → $\infty)$

Finding the General Solution
Let $A$ be a $3 \times 3$ matrix with eigenvalues $3,2,$ and $\frac{1}{2}$ , and corresponding eigenvectors and . if , find the general solution of the equation $x_{k+1}=Ax_k$ .

Analyzing the Long Term Behaviour
Explain the long term behaviour $(k$ → $\infty)$ of the equation $x_{k+1}=Ax_k$ , where

And where $c_1$ > $0$ and $c_2$ > $0$ .

Explain the long term behaviour $(k$ → $\infty)$ of the equation $x_{k+1}=Ax_k$ , where

And where $c_1$ > $0$ and $c_2$ > $0$

Find the matrix $A$ for the equation $x_{k+1}=Ax_k$ .

Suppose the eigenvalues of $A$ is $\lambda_1=1.0377$ and $\lambda_2=0.562303$ and the corresponding eigenvectors are and (assume $a,b,c,d$ > $0$ ). Find the general solution.

Assuming that $c_1,\;c_2$ > $0$ , explain the population of rabbits and eagles as $k$ → $\infty$ .