Gradient Systems and Poincare Index Theory Flashcards
Gradient System
Definition
-a dynamical system |x'=|F(|x) is called a gradient dynamical system if there exists a smooth function Φ(x), a potential, such that: |F(|x) = - ∇ (Φ(x)) -or in its components: F1(x1, ... , xn) = -∂Φ(x1,...,xn)/∂x1 F2(x1,...,xn) = -∂Φ(x1,...,xn)/∂x2 ... Fn(x1,...,xn) = -∂Φ(x1,...,xn)/∂xn
Gradient System Periodic Orbits Proposition
Statement
-a gradient does not admit periodic orbits
Gradient System Periodic Orbits Proposition
Proof
-let |x(t) be a non-constant solution of a gradient dynamical system:
|F(|x) = - ∇ (Φ(x))
-and:
φ(t) = Φ(|x(t))
-then for any t1>to :
φ(t1) - φ(to) = ∫ dΦ(x)/dt dt
-where the integral is taken between to and t1
= ∫ Σ ∂Φ/∂xi dxi/dt dt
-where the sum is taken from i=1 to i=n,
= ∫ F ∇Φ dt
= ∫ -∇Φ . ∇Φ dt < 0
-if there exists a periodic orbit |x(t+τ) = |x(t) with some period τ then:
φ(t+τ) - φ(t) =
Φ(|x(t+τ)) - Φ(|x((t)) = 0
-but this contradicts the above inequality so periodic orbits cannot exist
How to determine if a given system |x’=|F(|x)is gradient?
-if:
Fi = -∂Φ/∂xi, then:
∂Fi/∂xj = - ∂²Φ/∂xi∂xj = ∂Fj/∂xi
-for all i,j ϵ{1,...,N}
-then these Frobenius conditions are sufficient to prove that there exists a potential Φ(x) such that |F(|x) = - ∇ (Φ(x)), i.e. the system is gradientIdentifying Gradient Systems
N=1
-for N=1, every system is gradient with the potential:
Φ(x) = - ∫ |F(x) dx
-and we already know that in the one dimensional case there are no periodic orbits
Identifying Gradient Systems
N=2
-for a system of equations: x' = f(x,y) y' = g(x,y) -there is one condition: ∂f/∂y = ∂g/∂x -if this condition is satisfied, there exists a potential Φ(x,y) such that: f = -∂Φ/∂x g = -∂Φ/∂y
Identifying Gradient Systems
N=3
-in this case there are three Frobenius conditions which can be written in the form:
∇ x |F(|x) = 0
-if they are satisfied, then there exists and can be found a potential such that:
|F(|x) = - ∇ (Φ(x))
Identifying Gradient Systems
N≥3
-there are:
N(N-1)/2
-conditions which are necessary and sufficient for the existence of a potential
Gradient Systems and Lyapunov Functions
Remark
-let x* be a fixed point of a gradient system, then: Φ(x*) = Φo -let: V(|x) = Φ(|x) - Φo -then: i) V(x*) = 0 ii) if in a small vicinity of x*, V(x) is positive definite then V(x) is a Lyapunov function iii) dV/dt = - |∇Φ|² < 0
Poincare Index Theory
Description
- associated with any closed curve in the plane is an integer called the Poincare Index of the curve
- it measures the winding of the vector field
Poincare Index
Geometric Formula
-let Γ be a simple closed curve that does not pass through the any equilibrium points:
IΓ = Δφ/2π
-where I is the index and Γ is a subscript
Poincare Index
Formula Derivation
-let Γ be a simple closed curve that does not pass through any equilibrium points
-define φ(|x) to be the angle of the vector field (f(x,y) , g(x,y)) at any point |xϵΓ
-then:
tan(φ) = g(x,y) / f(x,y)
-now let |x go around Γ in an anticlockwise sense and define Δφ to be the change in φ as |x goes around Γ
-the Poincare index is the number of times the vector rotates, i.e.
I = Δφ/2π
Computing the Poincare Index
-the Poincare Index can be computed either by drawing vectors and counting the turns of the vector field or from the formula:
I = 1/2π ∮ dφ
= 1/2π ∮ [x’dy’ - y’dx’]/[x’²+y’²]
= ∮[f(∂g/∂x dx + ∂g/∂y dy) - g(∂f/∂x dx + ∂f/∂y dy)] / [f²+g²]
-using tan(φ) = y’/x’
y’=g and x’=f
Properties of the Poincare Index
i) if Γ’ is another curve obtained from Γ by a smooth deformation of Γ (without crossing any equilibrium) then: IΓ’ = IΓ (Γ subscript)
ii) if Γ does not enclose any equilibrium points, then IΓ=0
iii) if Γ is a periodic orbit, then IΓ=1
iv) the index is unchanged if (f,g) is replaced by (-f,-g)
Periodic Orbits and Equilibrium Points Corollary
-a periodic orbit must contain at least one equilibrium point
