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Dynamical Systems Analysis

LECTURE 5. Dynamical Systems Analysis. 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

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Dynamical Systems Analysis

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  1. LECTURE 5 Dynamical Systems Analysis

  2. 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 overtimegiven any current state (the future or long-termbehavior) before solving it numerically For instance, are there equilibrium states (physics) or fixed points (mathematics)? Are these statesstable or unstable?

  3. 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.

  4. Fixed points of First-order autonomous systems nonautonomous systems autonomous systems 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.

  5. E.g.: dx/dt = Ax(1 – x) withA=6.What are the fixed points? Set: dx/dt = 0 ie: 6x(1-x) = 0 so either x = 0 or x = 1 Therefore 2 fixed points, and how about the stability? Perturb the points and see what happens under the system dynamics… Use difference equation: x(t+h) = x(t) +h dx/dt from various different initial values of x

  6. Vector fields (t+h, x(t)+hdx/dt) (t, x(t)) 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

  7. Phase flow in 1-D phase space 1-D phase space hdx/dt

  8. First-order autonomous systems 1. Find fixed points: 2. Identify stability －using phase flow as above －using the gradient at the fixed points xkis stable xkis unstable

  9. Exercise 1: 1. Fixed points: x=0 2. Stable or not? If a<0, d(ax)/dx<0. stable If a>0, d(ax)/dx>0. unstable

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

  11. Second-order autonomous systems Suppose we have the following system: 1. Find the fixed points by setting: 2. Identify stability by Jacobian matrix (will not talk here) or 2-D phase space portrait

  12. Phase space portrait The 2-D space of possible initial conditions in which each solution follows a trajectory given by the vector field y (x(t+h),y(t+h)) =(x(t)+hdx/dt, y(t)+hdy/dt) (x(t),y(t)) x

  13. Example 1: A damped simple pendulum

  14. A damped simple pendulum

  15. Second-order autonomous systems First-order autonomous system in two variables Find the fixed points: (0, 0), (±π, 0), ……

  16. Phase space portrait of the damped simple pendulum (Created by AI Lehnen)

  17. Example 2: the Van der Pol oscillator

  18. Van der Pol oscillator The second term is not a constant like that in the damped simple pendulum with Second-order autonomous systems first-order autonomous system in two variables

  19. Phase space portrait with λ= 1. Find the fixed points: (0, 0). Stable or not?

  20. The Limit Cycle

  21. High order autonomous systems Suppose we have the following system (imagine a neural network …): We basically need a n-dimensional phase space.

  22. Homework • • 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 elucidatethe behaviour of • simple systems

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