Discrete dynamical systems

Discrete dynamical systems

Lessons

Assume that AA is diagonalizable, with nn linearly independent eigenvectors v1,v2,,vnv_1, v_2, \cdots , v_n, and corresponding eigenvalues λ1,λ2,λn\lambda _1, \lambda _2, \cdots \lambda _n. Then we can write an initial vector x0x_0 to be:
x0=c1v1+c2v2++cnvnx_0=c_1 v_1+c_2 v_2+ \cdots +c_n v_n

Let’s say we want to transform x0x_0 with matrix AA. Let’s call the transformed vector to be x1x_1. Then,
x1=Ax0=c1Av1+c2Av2++cnAvnx_1=Ax_0=c_1 Av_1+c_2 Av_2+\cdots+c_n Av_n
=c1λ1v1+c2λ2v2++cnλnvn=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 AkA\; k times. Then we can generalize this to be:
xk=c1(λ1)kv1+c2(λ2)kv2++cn(λn)kvnx_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 kk \infty.
  • 1.
    Discrete Dynamical Systems Overview:
    a)
    The Differential Equation xk+1=Axkx_{k+1}=Ax_k
    • Linear Transformation
    • Multiplying with A k times
    • Generalized formula
    • Why is this formula useful?

    b)
    Doing an Example
    • Calculate xkx_k
    • Long-term behaviour (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_k
    • Calculating the general solution
    • Long term behaviour (k(k ) \infty)


  • 2.
    Finding the General Solution
    Let AA be a 3×33 \times 3 matrix with eigenvalues 3,2,3,2, and 12\frac{1}{2}, and corresponding eigenvectors corresponding eigenvectors 1, 2 and corresponding eigenvectors 3. if initial vector x_0 , find the general solution of the equation xk+1=Axkx_{k+1}=Ax_k.

  • 3.
    Analyzing the Long Term Behaviour
    Explain the long term behaviour (k(k ) \infty) of the equation xk+1=Axkx_{k+1}=Ax_k, where
    Analyzing the Long Term Behaviour, Matrix A

    And Analyzing the Long Term Behaviour, initial vector x_0 where c1c_1 > 00 and c2c_2 > 00.

  • 4.
    Explain the long term behaviour (k(k ) \infty) of the equation xk+1=Axkx_{k+1}=Ax_k, where
    Analyzing the Long Term Behaviour, Matrix A

    And Analyzing the Long Term Behaviour, initial vector x_0 where c1c_1 > 00 and c2c_2 > 00

  • 5.
    Predator and Prey Model
    Let the eagle and rabbit population at time kk be denoted as initial vector k, where kk is the time in years, EkE_k is the number of eagles at time kk, and RkR_k is the number of rabbits at time kk (all measured in thousands). Suppose there are two equations describing the relationship between these two species:

    Ek+1=(.4)Ek+(.5)Rk E_{k+1}=(.4) E_k+(.5)R_k
    Rk+1=(.207)Ek+(1.2)Rk R_{k+1}=(-.207) E_k+(1.2) R_k
    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 AA for the equation xk+1=Axkx_{k+1}=Ax_k.

    c)
    Suppose the eigenvalues of AA is λ1=1.0377\lambda_1=1.0377 and λ2=0.562303\lambda_2=0.562303 and the corresponding eigenvectors are corresponding eigenvector v_1 and corresponding eigenvector v_2 (assume a,b,c,da,b,c,d > 00). Find the general solution.

    d)
    Assuming that c1,c2c_1,\;c_2 > 00, explain the population of rabbits and eagles as kk \infty.