Flashcards in Matrix Exponentials, Hyperbolic Fixed Points, Dulac's Negative Criterion and Lyapunov Functions Deck (28)

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1

## What type of equations does the matrix exponential method solve?

### -linear differential systems with constant coefficients

2

## Taylor Series of an Exponential Function in 1D

### e^(at) = 1 + at + 1/2! at² + 1/3! at³ + ...

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## Taylor Series of an Exponential Function of a Matrix

###
exp(At) = I + At + 1/2! A²t² + 1/3! A³t³ + ...

-where A is an nxn matrix and I is the identity matrix

-this series always converges and can be computed in a closed form

4

##
Proposition:

Let A be a square matrix with constant entries...

###
...then:

i) the exponential matrix X(t)=exp(At) satisfies the differential equation X'=AX

ii) any solution X(t) of the differential equation X'=AX satisfying the initial condition X(0)=I is given by X(t)=exp(At)

iii) solution of the initial value problem |x'=A|x, |x(0)=|xo is given by |x(t) = exp(At)*|xo

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## Properties of Matrix Exponentials

###
1) exp(0A) = I

2) if AB=BA, then:

exp(A)exp(B) = exp(A+B) = exp(B)exp(A)

3) if A = UBU^(-1) then, exp(UBU^(-1)) = Uexp(B)U^(-1)

4) if A is 2x2 matrix with first row α, 0 and second row 0,β, then exp(A) is a 2x2 matrix with entries first row: e^α, 0 and second row: 0, e^β

6

## What are the two cases for the matrix exponential method?

###
Case a -> λ1≠λ2

Case b -> λ1=λ2

7

##
Case B: λ1=λ2

Solution

###
-let (2x22) matrix A have eigenvalues λ1=λ2, then:

exp(At) = e^(λ1*t) * (I + t(A-λ1*I))

8

##
Proposition

λ1=λ2

###
-let A be a 2x2 matrix with such that λ1=λ2 (i.e. eigenvalues coincide)

-then:

(A-λ1*I)² = |0

-this is also true for nxn matrices with coninciding eigenvalues:

(A-λ1*I)^n = 0

9

##
Nilpotent

Definition

### -a matrix B such that B^n = |0 for some n∈ℕ is called a nilpotent matrix

10

##
Case B) λ1=λ2

Proof

###
A = A - λ1*I + λ1*I

-let B = A - λ1*I

A = B + λ1*I

-since I commutes with any matrix, [B,λ1*I]=0, B and λ1*I commute

exp(At) = exp(Bt + λ1*It) = exp(Bt)exp(λ1*It)

-expand the exponentials using the Taylor series:

exp(Bt) = I + Bt + B²t²/2! + ...., by B²=0 by the λ1=λ2 proposition

exp(Bt) = I + Bt + 0 + 0 + ... = I + Bt

exp(λ1*It) = exp(2x2) with entries tλ1, 0, 0, tλ1

= exp(2x2) with entries e^(tλ1), 0, 0, e^(tλ1) = e^(tλ1)*I

-sub back in:

exp(At) = (I + Bt) (e^(tλ1)I)

= e^(tλ1)*(I + Bt)

= e^(tλ1)*(I + t(A-λ1I))

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##
Case A) λ1≠λ2

Solution

###
exp(At) = U*(2x2)*U^(-1)

-where the 2x2 has entries e^(tλ1), 0, 0, e^(tλ2)

12

##
Case A) λ1≠λ2

Proof

###
-consider 2x2 matrix A with eigenvalues λ1, λ2 such that λ1≠λ2

-has eigenvalues |v1, |v2 which will be linearly independent since λ1≠λ2

-let U=(|v1, |v2)

AU = A(|v1,|v2) = (A|v1, A|v2) = (λ1|v1,λ2|v2) = (|v1,|v2)(2x2)

-where 2x2 has entries, λ1, 0, 0, λ2, so:

AU = U(2x2)

AUU^(-1) = A = U(2x2)U^(-1)

-take exponential:

exp(At) = exp(U(2x2)U^(-1)) where 2x2 now has entries tλ1, 0, 0, tλ2

exp(At) = exp(U(2x2)U^(-1)) = Uexp(2x2))U^(-1)

exp(At) = U(2x2)U^(-1)

-where (2x2) now has entries e^(tλ1), 0, 0, e^(tλ2)

13

## Behaviour at Hyperbolic vs Non-Hyperbolic Fixed Points

###
-at a non-hyperbolic point a small peterbation of the system can completely change the nature stability of the fixed poin

-at a hyperbolic fixed point this doesn't happen

14

##
Hyperbolic Fixed Point

Linear System Definition

###
-a fixed point at the origin of the linear system |x'=A|x is hyperbolic if each eigenvalue of A has non-zero real part

-otherwise it is called a non-hyperbolic fixed point

15

##
Hyperbolic Fixed Point

Non-Linear System Definition

### -a fixed point |x* of a non-linear system is hyperbolic if it is hyperbolic for the corresponding linearised system

16

## Hartman-Grobman Theorem

###
-the local phase portrait near a hyperbolic fixed point is 'topologically equivalent' to the phase portrait of the linearised system

-here 'topological equivalence' means there is a homomorphism that maps one phase portrait onto the other, i.e. trajectories map onto trajectories and the orientation is preserved

17

## Lotka-Volterra System

###
-Lotka-Volterra systems are used to model populations:

x' = x(a-bx-cy)

y' = y(d-ey-fx)

-where x and y are the populations of two different species

-a and d are the growth of the population from reproduction

-b and e are control of population size from amount of food, disease etc.

-c and f are competition terms from interaction between the two populations

-looking at the phase portrait, each point can be treated as an initial population and following the trajectory shows how the that population would evolve with time

18

##
Dulac's Negative Criterion

Theorem

###
let x'=f(x,y) and y'=g(x,y) be a continuously differentiable vector field in a simply connected region R⊆ℝ

-if there exists a continuously differentiable (in R) function h(x,y) such that:

∂h(x,y)f(x,y)/∂x + ∂h(x,y)g(x,y)/∂y

-is positive (or negative) throughout R, then there are no periodic orbits lying entirely within R

19

##
Dulac's Negative Criterion

Proof

### -proof by contradiction

20

##
Lyapunov Function

Definition

###
-suppose that x*ϵℝ^n is a fixed point of a dynamical system |x'=|F(|x)

-let R be an open neighbourhood of |x*, and R^ be the closure of R (i.e. R including the boundary)

-let V(x): R^->ℝ be a continuously differentiable function

-a function V(x) satisfying the following conditions

i) V(|x*) = 0

ii) V(x)>0 for all xϵR, x≠x*, (V is positive definite)

iii) V'<0 for all xϵR, x≠x*, (V' is negative definite)

-is called a Lyapunov function

21

##
Weak Lyapunov Function

Definition

###
-a function V(x) satisfying the following conditions:

i) V(|x*)=0

ii) V(|x)>0 for all xϵR, x≠x*, (V is positive definite)

iii) V'≤0 for all xϵR, x≠x*, (V is negative semi-definite)

-is called a weak Lyapunov function

22

## Implications of the Existence of a Lyapunov Function

###
1) there are no other fixed points except |x* in R^

2) for Lyapunov or weak Lyapunov, there is a region B⊆R of |x* such that all trajectories starting in B will stay in B, for Lyapunov (not weak) existence also implies that trajectories starting in B tend towards the fixed point |x*

3) there are no periodic orbits in R^

4) |x* is an asymptotically stable point or if V is a weak Lyapunov function, |x* is a stable point

23

##
Stable Fixed Point of a Dynamical System

Informal Definition

### -a fixed point |x*of a dynamical system |x'=|F(|x) is called stable if solutions with initial conditions sufficiently close to |x* exist for all t>0 and stay close to |x* at any t>0

24

##
Stable Fixed Point of a Dynamical System

Formal Definition

###
-a fixed point |x* of a dynamical system |x'=|F(|x) is called stable if:

-> for any ε>0 there exists 𝛿>0 such that all solutions |x(t) with initial conditions |x(0)=|xo, ||x0-|x*|0 and ||x(t)-|x*|

25

##
Asymptotically Stable Fixed Point of a Dynamical System

Definition

### -a fixed point is asymptotically stable if it is stable AND |x(t)->|x as t->∞

26

## Lyapunov Functions and Stability of Fixed Points

###
-the existence of a weak Lyapunov function guaranties the stability of the fixed point

-the existence of a Lyapunov function guaranties the asymptotic stability of the point

27

## Global Lyapunov Functions

###
-if a Lyapunov function V(x) exists for all |xϵℝ^n then it is a global Lyapunov function

-this means that the system has only one fixed point which is a global attractor

-i.e. every trajectory tends to this fixed point as t->∞

-and the existence of a global Lyapunov function rules out any periodic orbits

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