Gradient Systems and Poincare Index Theory Flashcards Preview

MATH2391 Non-Linear Differential Equations > Gradient Systems and Poincare Index Theory > Flashcards

Flashcards in Gradient Systems and Poincare Index Theory Deck (20)
Loading flashcards...
1

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

2

Gradient System Periodic Orbits Proposition
Statement

-a gradient does not admit periodic orbits

3

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

4

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 gradient

5

Identifying 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

6

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

7

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))

8

Identifying Gradient Systems
N≥3

-there are:
N(N-1)/2
-conditions which are necessary and sufficient for the existence of a potential

9

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

10

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

11

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

12

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

13

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

14

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)

15

Periodic Orbits and Equilibrium Points Corollary

-a periodic orbit must contain at least one equilibrium point

16

Poincare Index of an Isolated Fixed Point

-define the index of an isolated equilibrium point to be the index of any simple closed curve that encircle that equilibrium point and no others, the index is independent of the choice of curve
-the index of a node, focus, star or centre is +1 (for stable or unstable)
-the index of a saddle is -1

17

Using Equilibrium Points to Calculate the Poincare Index

-the index of ay simple closed curve is the sum of the indices of the equilibrium points it encloses

18

Isolated Fixed Point
Definition

-a fixed point (x*,y*) is isolated if there exists an open neighbourhood U with (x*,y*)ϵU such that U does not contain any other fixed points
-e.g. a fixed point in a line of fixed point is not isolated as no such U exists

19

Poincare-Bendixson Theorem

Suppose that:
i) |x'=|f(|x) is a continuously differentiable vector field on an open set containing R
ii) R does not contain any equilibrium points
iii) there exists a trajectory C that is confined in R ,i.e. it starts in R and remains there for all t
THEN
-either C is a periodic orbit or C spirals towards a periodic orbit
-either way, R CONTAINS A PERIODIC ORBIT

20

What is the easiest way to show that there exists a curve C that obeys the Poincare-Theorem?

-to show that R is a trapping region, i.e. that all trajectories that enter R do not leave
-remember that R has a hole in the middle for fixed points
-the vector field on the boundary of R must point into the region R (or be tangent to the boundary) at every point on the boundary