Matrix Exponentials, Hyperbolic Fixed Points, Dulac's Negative Criterion and Lyapunov Functions Flashcards
What type of equations does the matrix exponential method solve?
-linear differential systems with constant coefficients
Taylor Series of an Exponential Function in 1D
e^(at) = 1 + at + 1/2! at² + 1/3! at³ + …
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
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
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
What are the two cases for the matrix exponential method?
Case a -> λ1≠λ2
Case b -> λ1=λ2
Case B: λ1=λ2
Solution
-let (2x22) matrix A have eigenvalues λ1=λ2, then:
exp(At) = e^(λ1t) * (I + t(A-λ1I))
Proposition
λ1=λ2
-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-λ1I)^n = 0
Nilpotent
Definition
-a matrix B such that B^n = |0 for some n∈ℕ is called a nilpotent matrix
Case B) λ1=λ2
Proof
A = A - λ1I + λ1I
-let B = A - λ1I
A = B + λ1I
-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))
Case A) λ1≠λ2
Solution
exp(At) = U(2x2)U^(-1)
-where the 2x2 has entries e^(tλ1), 0, 0, e^(tλ2)
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)
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
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
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
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
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
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
Dulac’s Negative Criterion
Proof
-proof by contradiction
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
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
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
Stable Fixed Point of a Dynamical System
Informal Definition
-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
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|
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->∞
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
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
How to find a Lyapunov function
- there is no set formula
- good candidates for the fixed point |x=0 are:
i) V(x) = Σanxn²
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
