程序代做CS代考 MAT332 – cscodehelp代写

MAT332
Introduction to Nonlinear Dynamical Systems and Chaos
Lecture 2
Qun WANG
University of Toronto Mississauga
September 14, 2021
(UTM)
MAT332 Intro Nonlinear Dyn and 14, 2021
1 / 19

Review
In general, which of the following systems might be more complicated to study:
􏰧 A linear system;
􏰧 A nonlinear system with degree of freedom equal to 2;
􏰧 A nonlinear system with degree of freedom equal to 10.
What is the phase space of the following mechanical system with x = (x1, x2) ∈ R2, governed by the equation
x ̈1(t) = tan (x1(t) − x2(t)); x ̈1(t) = tan (x2(t) − x1(t)).
Is it autonomous or non-autonomous? Can you find a fixed point of this system? What is the phase space?
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 2 / 19

Learning Objective
For today’s lecture, students are expected to:
Objective
Know the basic result on the existence and uniqueness of linear dynamical systems (without proof);
Understand the notion of fundamental set of solutions of a linear dynamical system;
Apply Wronskian and Liouville formula for verification of fundamental set of solutions.
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 3 / 19

Linear Dynamical System
Recall that:
Definition (Linear System)
Consider the dynamical system x ̇(t) = f(t,x(t)). If f consists of only linear functions (on x), then the system is called linear. Let A(t) be its coefficient matrix, the system can be denoted by
x ̇(t) = A(t)x(t) (1) For each given t ∈ R, x(t) ∈ Rn is considered as a column vector,
and A(t) ∈ Rn×n is considered as a matrix.
Note that we are working on a general case: A(t) can depends on t,
hence the system could be non-autonomous.
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 4 / 19

Linear Dynamical System
Example (Euler Differential Equation)
Consider the 2nd order differential equation with y ∈ R and t ̸= 0: t2y ̈ + 2ty ̇ − 2y = 0
Show that it is a linear system with nonconstant coefficient (i.e., the coefficient matrix depends on t)
Solution
(2)
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021
5 / 19

Solutions of Linear Dynamical System
Definition (Solution)
A solution of the equation
x ̇(t) = A(t)x(t) (3) with the initial condition x(t0) = x0, is a function x∗(t) defined on some
open interval {t0} ⊂ I ⊂ R s.t.
∀t ∈ I, x∗(t) satisfies the equation ;
x∗ satisfies the initial condition, i.e. x∗(t0) = x0
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 6 / 19

Example
Recall that just now we have introduced the linear dynamical system coming from Euler differential equation:
x ̇ 1 = x 2 ;
x ̇ 2 = 2 x 1 − 2 x 2 .
t2 t
􏰥0􏰦 ∗ 􏰥t􏰦
initial condition x(0) = 1 . Show that x (t) = 1 is a solution. Solution
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 7 / 19

Solutions of Linear Dynamical System
Once we defined our solution, the next natural question we would like to ask, is the existence and uniqueness. For linear dynamical systems, we have the following theorem:
Theorem
Suppose that the coefficient matrix A(t) = (aij)1≤i,j≤n satisfies that Each aij is a continuous function of t
∃c ∈ R s.t. |aij(t)| < c,∀1 ≤ i ≤ j ≤ n,t ∈ R. Then given any initial condition, the system has a unique solution, and the solution exists for all t ∈ R. Example Does Euler differential equation satisfy the above hypothesis? (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 8 / 19 Fundamental Set of Solutions Now the following proposition permits us to generate new solutions from old ones. Proposition Suppose that x and y both solve the equation with initial conditions x(0) = x0, y(0) = y0 respectively. Then for any real number c1, c2 ∈ R, the linear combination. z = c1x + c2y solves the equation with initial condition z(0) = c1x0 + c2y0 as well. This implies that if we define FA = {x(t) : solutions of the linear system x ̇(t) = A(t)x(t)} Then FA is a linear space. (4) (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 9 / 19 Fundamental Set of Solutions Now that FA = {x(t) : solutions of the linear system x ̇(t) = A(t)x(t)} is a vector space, we want to study its basis. The initial condition can be specified at t = 0 (Why?). Now let e1, e2, ...en be a basis in Rn. We consider the solutions: xi(t), with initial condition xi(0) = ei, 1 ≤ i ≤ n (5) We show that {xi(t)}1≤i≤n is a basis of FA. In other words, we need to show that 􏰧 {xi (t)}1≤i≤n are (linearly) independent. 􏰧 {xi(t)}1≤i≤n span FA. (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 10 / 19 Fundamental Set of Solutions Linear Independence: Proposition Suppose that for c1, c2, ..., cn ∈ R, one has that c1x1(t) + c2x2(t) + ... + cnxn(t) = 0, thenc1 =c2 =...=cn =0. Proof. ∀t ∈ R (6) (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 11 / 19 Fundamental Set of Solutions Span the whole space Proposition Suppose that x(t) ∈ FA is a solution of the system with initial condition x(0) = x0, then there exists c1, c2, ..., cn ∈ R, s.t. Proof. c1x1(t) + c2x2(t) + ... + cnxn(t) = x(t), ∀t ∈ R (7) (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 12 / 19 Fundamental Set of Solutions Definition (Fundamental Set of Solutions) Consider a linear system x ̇(t) = A(t)x(t) (8) in Rn. Then a basis {xi(t)}1≤i≤n of FA is called a fundamental set of solutions. Is the fundamental set of solutions unique for a given linear system? (Hint: is the basis of a vector space unique? ) (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 13 / 19 Fundamental Set of Solutions We introduce a convenient way to represent the fundamental set of solutions: Definition (Matrix Solution) Any M(t) ∈ Rn×k satisfying is called a matrix solution. Example d M(t) = A(t)M(t) dt (9) If x1, x2, ...xk satisfies the differential equation, then  M=x1 x2 ...xk is a matrix solution. (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 14 / 19 Fundamental Set of Solutions Theorem Let x1(t),x2(t),...xk(t) in Rn. The following are equivalent: 1 Each xi(t) satisfies the equation x ̇i(t) = A(t)xi(t); 2 The matrix M ∈ Rn×k  M=x1 x2 ...xk is a matrix solution. 3 For any constant vector c = (c1,c2,...,ck)T, M(t)c is a solution (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 15 / 19 Proof. (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 16 / 19 With the n × n matrix solution in hand, we can easily check the independence of its columns (which are solutions of the linear dynamical systems) by calculating the Wronskian. Definition (Wronskian of the matrix solutions) Given M(t) ∈ Rn×n a matrix solution, the Wronskian, W (t), is defined as W (t) = det(M(t)) W(t) is a function, for given t it is a number. (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 17 / 19 Wronskian of a Matrix Solution Theorem (Liouville Formula) Suppose that M(t) ∈ Rn×n is a matrix solution for a linear system x ̇(t) = A(t)x(t), and W(t) = det(M(t)) be its Wronskian. Then W(t) satisfies the differential equation d W(t) = tr(A(t))W(t) (10) dt , which then implies that W(t) = W(t0)exp( 􏰣t t0 tr(A(s))ds) (11) In particular this implies that if the W (t0) ̸= 0, then W (t) ̸= 0 for all t (whenever M(t) is well defined on t). (UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 18 / 19 Wronskian of a Matrix Solution Example Consider the Euler differential system x ̇ 1 = x 2 ; x ̇ 2 = 2 x 1 − 2 x 2 . t2 t Find a fundamental set of solutions for the system when t > 0.
(UTM) MAT332 Intro Nonlinear Dyn and 14, 2021 19 / 19