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MATH 33B Lecture Notes - Lecture 18: Eigenvalues And Eigenvectors

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rubyporcupine828
17 Nov 2016
School
UCLA
Department
Mathematics
Course
MATH 33B
Professor
Visan

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We can write this as a first-order equation for a vector. (cid:883) Note: (cid:1876) (cid:4666)(cid:4667)=((cid:1876)(cid:2869) (cid:4666)(cid:4667)(cid:1709)(cid:1876) (cid:4666)(cid:4667)) (cid:1827)(cid:1876)=(cid:3438)(cid:2869)(cid:2869) (cid:2869)(cid:2870) (cid:2869) (cid:2869) (cid:2870) )(cid:3438)(cid:1876)(cid:2869)(cid:1876)(cid:2870)(cid:1709)(cid:1876))=(cid:3438)(cid:2869)(cid:2869)(cid:1876)(cid:2869)+(cid:2869)(cid:2870)(cid:1876)(cid:2870)+(cid:1710)+(cid:2869)(cid:1876) (cid:2870)(cid:2869) (cid:2870)(cid:2870) (cid:2870) (cid:2870)(cid:2869)(cid:1876)(cid:2869)+(cid:2870)(cid:2870)(cid:1876)(cid:2870)+(cid:1710)+(cid:2870)(cid:1876) (cid:2869)(cid:1876)(cid:2869)+(cid:2870)(cid:1876)(cid:2870)+(cid:1710)+(cid:1876))=(cid:1827)(cid:2869)(cid:1876)(cid:2869)+(cid:1710)+(cid:1827)(cid:1876) (cid:1709) (cid:1709) (cid:1709) where (cid:1827)(cid:3037)=(cid:4684)(cid:1827)(cid:2869)(cid:3037)(cid:1709)(cid:1827)(cid:3037)(cid:4685) is the th column of (cid:1827) A solution is a continuously differentiable vector-valued function (cid:1876): such that (cid:1876) (cid:4666)(cid:4667)=(cid:1827)(cid:1876)(cid:4666)(cid:4667) Note that if (cid:1876) and (cid:1877) are solutions and (cid:1855)(cid:2869),(cid:1855)(cid:2870) , then (cid:1855)(cid:2869)(cid:1876)+(cid:1855)(cid:2870)(cid:1877) is a solution. (cid:4666)(cid:1855)(cid:2869)(cid:1876)+(cid:1855)(cid:2870)(cid:1877)(cid:4667) =(cid:1855)(cid:2869)(cid:1876) +(cid:1855)(cid:2870)(cid:1877) =(cid:1855)(cid:2869)(cid:1827)(cid:1876)+(cid:1855)(cid:2870)(cid:1827)(cid:1877) For any (cid:1876)(cid:2868) , there exists a unique solution to (cid:1876) =(cid:1827)(cid:1876) with (cid:1876)(cid:4666)(cid:2868)(cid:4667)=(cid:1876)(cid:2868) where (cid:2868) . Assume that (cid:1877)(cid:2869), ,(cid:1877) are linearly independent solutions to (cid:1876) =(cid:1827)(cid:1876) Then all solutions to the system can be expressed as linear combinations of (cid:1877)(cid:2869), ,(cid:1877), that is, there exist (cid:1855)(cid:2869), ,(cid:1855) such that (cid:1876)=(cid:1855)(cid:2869)(cid:1877)(cid:2869)+(cid:1710)+(cid:1855)(cid:1877). (existence and uniqueness) (cid:1709) Can be written as a linear system of dimension . Indeed, let (cid:1876)(cid:2869)=(cid:1876),(cid:1876)(cid:2870)=(cid:1876) ,(cid:1876)(cid:2871)=(cid:1876) , ,(cid:1876)=(cid:1876)(cid:4666) (cid:2869)(cid:4667) and let (cid:1876)=((cid:1876)(cid:2869)(cid:1709)(cid:1876)) (cid:1876)(cid:2870)(cid:1876)(cid:2871)(cid:1709) (cid:1856)(cid:1856)(cid:1876)=(cid:3438)(cid:1876)(cid:2869) (cid:1876)(cid:2870) (cid:1709)(cid:1876) )=(cid:3438)(cid:1876) (cid:1876) (cid:1709)(cid:1876)(cid:4666)(cid:4667))=(cid:3438)

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Related Questions

A classical model in mathematical ecology is the Lotka–Volterra predator-prey model. Consider a simple ecosystem consisting of rabbits that have an infinite supply of food and foxes that prey on the rabbits for their food. This is modeled by a pair of nonlinear, first-order differential equations:

where t is time, r(t) is the number of rabbits, f(t) is the number of foxes, and α is a positive constant. If α = 0, the two populations do not interact, the rabbits do what rabbits do best, and the foxes die off from starvation. If α > 0, the foxes encounter the rabbits with a probability that is proportional to the product of their numbers. Such an encounter results in a decrease in the number of rabbits and (for less obvious reasons) an increase in the number of foxes. The solutions to this nonlinear system cannot be expressed in terms of other known functions; the equations must be solved numerically. It turns out that the solutions are always periodic, with a period that depends on the initial conditions. In other words, for any r(0) and f(0), there is a value t = tp when both populations return to their original values. Consequently, for all t,

(a) Compute the solution with r0 = 300, f0 = 150, and α = 0.01. You should find that tp is close to 5. Make two plots, one of r and f as functions of t and one a phase plane plot with r as one axis and f as the other.

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(d) The point (r0, f0) = (1/α, 2/α) is a stable equilibrium point. If the populations have these initial values, they do not change. If the initial populations are close to these values, they do not change very much. Let u(t) = r(t)−1/α and v(t) = f(t) − 2/α. The functions u(t) and v(t) satisfy another nonlinear system of differential equations, but if the uv terms are ignored, the system becomes linear. What is this linear system? What is the period of its periodic solutions?

Hi, I need the full Matlab code and answers for all parts of the problem. Thanks!