# Linear Algebra and Its Applications Linear Algebra Lecturer: Xia Liao Email: [email protected] Homepage: https://hirzebruch.github.io/main.html

Time: every Tuesday 1-2, Thursday 3-4 Discrete dynamic systems The

goal in this section is to study the sequence of vectors where is given and the sequence is defined by recursion Section 5.5 and 5.6 focus on the situation that is a matrix.

Consider the characteristic equation of . It is Since is a matrix, the characteristic equation is of the form where . The solutions of this equation are eigenvalues of .

4 cases of 2-dimensional discrete dynamic system s The book is discussing 4 possibilities according to the eig envalues of . Of course these are not all possibilities.

1. has 2 real eigenvalues . 2. has 2 complex eigenvalues .

0< 1 <1, 0< 2 <1 The general solution of this type of problems were discussed on page 300 exam

ple 5. There are 2 steps. Step1: find the eigenvalues () and eigenvectors () of . Write the initial value as a linear combination of and . Step2: We obtain the solution immediately.

The reason for the parabolic shapes of the traject ories The

vector So its coordinates are . Equivalently, . But , so we have Or Therefore the shape is a parabola.

1 >1, 2>1 The origin as a repeller

Ex: try to explain the parabolic shape of the trajectory 1 >1, 0< 2 <1

The origin as a saddle point The eigenvectors generate the asymptotes

A predator prey system Predator: Owl, Prey: Rat Notation: is the number of owls in the kth year. is the number of r

ats in the kth year. We need to determine the evolution of the system, starting with . T he method appeared in p300 example 3. It can also be solved usin

g the method of p304 example 2. Ex: Analyze the asymptotic behavior of the solution. Also demonstrate the meaning of the asymptotic solution in th

e original ecosystem. Complex eigenvalues There are also several subcases. We briefly discuss 2 cas

es. 1. If the norm of the complex eigenvalues are 1, then the trajectory of the discrete dynamic system is a closed e llipse.

2. If the norm of the complex eigenvalues are less than 1 , then the origin is an attractor. Norm=1

Page318, example 2,3. The eigenvalues are as the book computes.

Norm=1, trajectory Norm<1, the origin as an attractor P329: example 6

The extinction of the spotted owl The mathematical model:

Looking for eigenvalues: Writing out the general solution and analyze the asymptotic behavior of the solu tions:

Model revision: the survivial HW

The homework will not be collected on Thursday, but do not skip doing it at home by yourself. Impose self-mast ery! P331: 1,2

P332: read the solution of 3-6