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Discrete dynamical systems
Lessons
Assume that A is diagonalizable, with n linearly independent eigenvectors v1,v2,⋯,vn, and corresponding eigenvalues λ1,λ2,⋯λn. Then we can write an initial vector x0 to be:
x0=c1v1+c2v2+⋯+cnvn
Let's say we want to transform x0 with matrix A. Let's call the transformed vector to be x1. Then,
x1=Ax0=c1Av1+c2Av2+⋯+cnAvn
=c1λ1v1+c2λ2v2+⋯+cnλnvn
Let's say we want to keep transforming it with matrix Ak times. Then we can generalize this to be:
xk=c1(λ1)kv1+c2(λ2)kv2+⋯+cn(λn)kvn
This is useful because we get to know the behaviour of this equation when k→∞.
Let's say we want to transform x0 with matrix A. Let's call the transformed vector to be x1. Then,
=c1λ1v1+c2λ2v2+⋯+cnλnvn
Let's say we want to keep transforming it with matrix Ak times. Then we can generalize this to be:
This is useful because we get to know the behaviour of this equation when k→∞.
- IntroductionDiscrete Dynamical Systems Overview:a)The Differential Equation xk+1=Axk
• Linear Transformation
• Multiplying with A k times
• Generalized formula
• Why is this formula useful?b)Doing an Example
• Calculate xk
• Long-term behaviour (k→∞)c)Application: Predator and Prey Model
• Analyzing the predator and prey equations
• Converting to the equation xk+1=Axk
• Calculating the general solution
• Long term behaviour (k→∞) - 1.Finding the General Solution
Let A be a 3×3 matrix with eigenvalues 3,2, and 21, and corresponding eigenvectorsand
. if
, find the general solution of the equation xk+1=Axk.
- 2.Analyzing the Long Term Behaviour
Explain the long term behaviour (k→∞) of the equation xk+1=Axk, where
Andwhere c1 > 0 and c2 > 0.
- 3.Explain the long term behaviour (k→∞) of the equation xk+1=Axk, where
Andwhere c1 > 0 and c2 > 0
- 4.Predator and Prey Model
Let the eagle and rabbit population at time k be denoted as, where k is the time in years, Ek is the number of eagles at time k, and Rk is the number of rabbits at time k (all measured in thousands). Suppose there are two equations describing the relationship between these two species:
Ek+1=(.4)Ek+(.5)Rk
Rk+1=(−.207)Ek+(1.2)Rka)What happens to the rabbits if there are no eagles? What happens to the eagles if there are no rabbits?b)Find the matrix A for the equation xk+1=Axk.c)Suppose the eigenvalues of A is λ1=1.0377 and λ2=0.562303 and the corresponding eigenvectors areand
(assume a,b,c,d > 0). Find the general solution.
d)Assuming that c1,c2 > 0, explain the population of rabbits and eagles as k→∞.