Matrix Exponentials, Hyperbolic Fixed Points, Dulac's Negative Criterion and Lyapunov Functions Flashcards

1
Q

What type of equations does the matrix exponential method solve?

A

-linear differential systems with constant coefficients

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2
Q

Taylor Series of an Exponential Function in 1D

A

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

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3
Q

Taylor Series of an Exponential Function of a Matrix

A

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
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4
Q

Proposition:

Let A be a square matrix with constant entries…

A

…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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5
Q

Properties of Matrix Exponentials

A

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

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6
Q

What are the two cases for the matrix exponential method?

A

Case a -> λ1≠λ2

Case b -> λ1=λ2

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7
Q

Case B: λ1=λ2

Solution

A

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

exp(At) = e^(λ1t) * (I + t(A-λ1I))

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8
Q

Proposition

λ1=λ2

A

-let A be a 2x2 matrix with such that λ1=λ2 (i.e. eigenvalues coincide)
-then:
(A-λ1I)² = |0
-this is also true for nxn matrices with coninciding eigenvalues:
(A-λ1
I)^n = 0

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9
Q

Nilpotent

Definition

A

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

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10
Q

Case B) λ1=λ2

Proof

A

A = A - λ1I + λ1I
-let B = A - λ1I
A = B + λ1
I
-since I commutes with any matrix, [B,λ1I]=0, B and λ1I commute
exp(At) = exp(Bt + λ1It) = exp(Bt)exp(λ1It)
-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(λ1It) = 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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11
Q

Case A) λ1≠λ2

Solution

A

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

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

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12
Q

Case A) λ1≠λ2

Proof

A

-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)

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13
Q

Behaviour at Hyperbolic vs Non-Hyperbolic Fixed Points

A
  • 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
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14
Q

Hyperbolic Fixed Point

Linear System Definition

A
  • 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
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15
Q

Hyperbolic Fixed Point

Non-Linear System Definition

A

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

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16
Q

Hartman-Grobman Theorem

A
  • 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
Q

Lotka-Volterra System

A

-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
Q

Dulac’s Negative Criterion

Theorem

A

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
Q

Dulac’s Negative Criterion

Proof

A

-proof by contradiction

20
Q

Lyapunov Function

Definition

A
  • 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
Q

Weak Lyapunov Function

Definition

A
  • 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
Q

Implications of the Existence of a Lyapunov Function

A

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
Q

Stable Fixed Point of a Dynamical System

Informal Definition

A

-a fixed point |xof 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
Q

Stable Fixed Point of a Dynamical System

Formal Definition

A
  • 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
Q

Asymptotically Stable Fixed Point of a Dynamical System

Definition

A

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

26
Q

Lyapunov Functions and Stability of Fixed Points

A
  • 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
Q

Global Lyapunov Functions

A
  • 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
28
Q

How to find a Lyapunov function

A
  • there is no set formula
  • good candidates for the fixed point |x=0 are:
    i) V(x) = Σan
    xn²
    ii) V(x) = Σ an*xn^(2mn)
  • all n are subscripts, sums taken between n=1 and n=N, mnϵℕ
  • for some choices of positive constants an and positive integers an