Dynamical Systems Analysis

LECTURE 5

Why?

A dynamical system (e.g. a neuron or a neural system) is usually described by a set of nonlinear differential equations

What is ‘analysis’ here?

To determine how the system behaves over time given any current state (the future or long-term behavior ) before solving it numerically

For instance, are there equilibrium states (physics) or fixed

points (mathematics)? Are these states stable or unstable?

Equilibria of a dynamical system

Coin balanced on a table － How many equilibria? (Face Up, Face Down, Edge) － Is it stable?

A ball in the track:

It’s either on top of a hill or at the bottom of the track. To find out which, push it (perturb it), and see if it comes back.

Fixed points of First-order autonomous systems

0) ,( xfdt

dx

) , ,( txfxdt

dx

By a fixed point we mean that x doesn’t change as time increases, i.e.:

So, to find fixed points just solve above equation.

) ,( xfdt

dx

autonomous systemsnonautonomous systems

E.g.: dx/dt = Ax(1 – x) with A=6. What are the fixed points?

Set: dx/dt = 0 ie: 6x(1-x) = 0so either x = 0 or x = 1

Use difference equation: x(t+h) = x(t) +h dx/dt from various different initial values of x

Therefore 2 fixed points, and how about the stability?

Perturb the points and see what happens under the system

dynamics…

Vector fields

0 0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t

x

(t, x(t))

(t+h, x(t)+hdx/dt)

So x=0 is unstable and x=1 seems to be stable. Change the value of parameter A to see the influence of parameters on systems

Phase flow in 1-D phase space

)1( xAxxdt

dx

hdx/dt

1-D phase space

dt

dx

First-order autonomous systems

0

0) ,(

dx

xdf k

) ,( xfdt

dx

1. Find fixed points:

2. Identify stability － using phase flow as above － using the gradient at the fixed points

xk is stable

xk is unstable

0),( kxf

Exercise 1:

adx

axd

)(

1. Fixed points: x=0

2. Stable or not?

If a<0, d(ax)/dx<0. stable

axdt

dx

If a>0, d(ax)/dx>0. unstable

Exercise 2: 3xxdt

dx

13)( 2

3

xdx

xxd

1. Fixed points: x=0, +1, -1 2. Stable or not?

Second-order autonomous systems

),(

),(

yxgydt

dy

yxfxdt

dx

Suppose we have the following system:

1. Find the fixed points by setting:

0),(

0),(

kk

kk

yxg

yxf

2. Identify stability by Jacobian matrix (will not talk here) or 2-D phase space portrait

Phase space portrait

The 2-D space of possible initial conditions in which each solution follows a trajectory given by the vector field

(x(t+h),y(t+h)) =(x(t)+hdx/dt, y(t)+hdy/dt)

(x(t),y(t))

x

y

),(

),(

yxgydt

dy

yxfxdt

dx

Example 1: A damped simple pendulum

A damped simple pendulum

)0(

0)sin(2

2

adt

da

dt

d

Second-order autonomous systems

First-order autonomous system in two variables

)sin(xaydt

dy

ydt

dx

dt

dyx

,

Find the fixed points: (0, 0), (±π, 0), ……

)0(

0)sin(2

2

adt

da

dt

d

Phase space portrait of the damped simple pendulum

)sin(4.0 xydt

dy

ydt

dx

(Created by AI Lehnen)

Example 2: the Van der Pol oscillator

Van der Pol oscillator

0)1( 22

2

xdt

dxx

td

xd

The second term is not a constant like that in the damped simple pendulum with )0( ,0)sin( aa

Second-order autonomous systems

first-order autonomous system in two variables

xyxdt

dy

ydt

dx

)1( 2

dt

dxy

Find the fixed points: (0, 0). Stable or not?

xyxdt

dy

ydt

dx

)1( 2

Phase space portrait with λ = 1.

The Limit Cycle

High order autonomous systems

) ..., ,,(

......

) ..., ,,(

) ..., ,,(

21

21222

21111

nnnn

n

n

xxxfxdt

dx

xxxfxdt

dx

xxxfxdt

dx

Suppose we have the following system (imagine a neural network …):

We basically need a n-dimensional phase space.

• Find fixed points of a simple system

• Classify fixed points as stable/unstable. Especially, use graphical methods (vector field plots in phase space) to analyze and elucidate the behaviour of simple systems

Homework