C24: Dynamical Systems

Lectures: Mark Cannon

Slides: Sina Ober-Blöbaum & Ron DanielLecture Notes: Antonis Papachristodoulou

Department of Engineering Science University of Oxford

Dynamical Systems: Lecture 1 1

Lecture 1: Introduc.on• We will study the topological properties of solutions of ordinary

differential equations without solving them

• Essentially about the geometry of the paths describing the time evolution of solutions and how such paths can be thought of as lying on ‘surfaces’

• Intimately associated with the idea of State (or Phase Space) – how solutions are related to a ‘state’

2Dynamical Systems: Lecture 1

Course summary

1. Introduction to dynamical systems

2. Phase space and equilibria

3. The stable, unstable and centre subspaces

4. Lyapunov functions, gradient and Hamiltonian systems. Vector fields possessing an integral

5. Invariance. La Salle’s theorem. Domain of attraction

6. Limit sets, attractors, orbits, limit cycles, Poincaré maps

7. Saddle-node, transcritical, pitchfork and Hopf bifurcations

8. Logistic map, fractals and Chaos. Lorenz equations

Dynamical Systems: Lecture 1 3

C24: Dynamical Systems• 8 lectures:– 11am on Thurs & Fri, weeks 5-8, LR2

• Examples class 1 (lectures 1-4):– Thu, week 8, 2-3pm or 3-4pm, LR4– Fri, week 8, 2-3pm or 3-4pm, LR5

• Examples class 2 (lectures 5-8):– Mon & Tue, week 1 Hilary 2019

• Revision class in Trinity • Lecture notes + slides available on WebLearn & markcannon.github.io

Dynamical Systems: Lecture 1 4

Examples of Dynamical Systems

• Single species growth: the logistic equation!"!# = %" 1 − "

(": population at time #% > 0: birth rate (: carrying capacity.Solution is lengthy! (see lecture notes):

" # = ,(-./( + ,-./

Dynamical Systems: Lecture 1 5

Examples of Dynamical Systems

• What does the analytic solution tell us? Is it very informative?

• What happens if ! 0 = 0. What does this mean?

• What happens when $®∞? (does !® &?)

• Can we analyse the solution properties without solving the equation?

• Try to introduce geometry into the problem.

Dynamical Systems: Lecture 1 6

Phase space

• Any !th order differential equation in a single unknown variable "($)can be written as ! coupled first order differential equations in !unknown variables "1($), "2($) … . "!($).

• Each variable defines a coordinate in phase space.

• Solutions are curves (or trajectories) in phase space determined by the initial conditions.

Dynamical Systems: Lecture 1 7

Names of Phase Spaces

• If ! = 1 we have a Phase Line.

• If ! = 2 we have a Phase Plane.

• If ! > 2 we have a general Phase Space.

• Collections of similar trajectories can form surfaces, sometimes called solution manifolds (a fancy name for a smooth surface).

• You encountered phase space in P1 ‘Mathematical Modelling’!

Dynamical Systems: Lecture 1 8

The single species revisited

• There are special points in Phase Space where the solution remains stationary, i.e. !"!# = 0.

• The single species equation is first order, & = 1, and the special points are when

() 1 − )+ = 0 ⟺ ) = 0 -. ) = +

• These special points are called equilibria.• As & = 1, the solution trajectories lie on a single Phase Line.

Dynamical Systems: Lecture 1 9

Consider !"!# = %& 1 − ") as a func.on of &:

& < 0 ⟹ -&-. < 0

0 < & < / ⟹ -&-. > 0

& > / ⟹ -&-. < 0

Dynamical Systems: Lecture 1 10

& = 0 & = /&

The resulting phase portrait

Stable and unstable equilibria

• All points near to ! = 0 move away from this equilibrium – it is unstable.

• All points near to ! = $ more towards this equilibrium – it is stable.

• It is not possible to go from ! < $ to ! > $ without ! = $ at some point – when it stops! Thus there is no overshoot at ! = $.

Dynamical Systems: Lecture 1 11

! = 0 ! = $!

Dynamical Systems: Lecture 1 12

The solu7on as a func7on of 7me

The damped simple pendulum

!"$̈ = −!'sin$ − +"$̇There are two states: let -. = $ and -/ = 012

034-.45 = -/4-/45 = −'sin -." − +

!-/

Dynamical Systems: Lecture 1 13

q

damping +

"

!'

The equilibria (both !"#!$ = 0 and !"'!$ = 0 ) are when () = 0 and sin (- = 0

Dynamical Systems: Lecture 1 14

-8 -6 -4 -2 0 2 4 6 8-8

-6

-4

-2

0

2

4

6

8The phase plane of the pendulum equation, l = 1, k = 1, m = 1, g = 10

x1, q

x2, dq/

dt

Glycolytic oscillations

• Involves turning glucose into energy compounds such as ATP within a cell:

"̇ = −" + &' + "(''̇ = ) − &' − "('

variables " and ' are concentrations of two intermediaries

Dynamical Systems: Lecture 1 15

Dynamical Systems: Lecture 1 16

System behaviour for one parameter set but with two different initial conditions

Glycolytic oscillations

The double pendulum

Dynamical Systems: Lecture 1 17

q1"#

q2

"

#

0 2 4 6 8 10 12 14 16 18 20-20

0

20

40

time

q 1, q

2

0 2 4 6 8 10 12 14 16 18 20-20

-10

0

10

time

q 1, q

2

Time evolution of angles in double pendulum

The two responses start at %1 = %2 = 2p/3, zero velocity, but the bottom adds 0.01 radians to %1 for its initial condition. Why are they so different?

The Mandelbrot set• An iterative equation:

!"#$ = !"& + (!" and ( are complex. If !) = 0, for which values of ( does |!"| remain bounded?

Dynamical Systems: Lecture 1 18

the set for general complex (effect of varying , with ( = , 1 + .

Our Strategy

• We will be studying equilibrium points of differential equations.• The nature of equilibria are largely defined by their local

linearisations.• We then study the geometry and topology (connectedness) of

regions around equilibria in phase space.• We reason about the nature of the flows through these regions.• To begin we need to understand the geometry of local

linearisations – revisit eigenvalues and eigenvectors of matrices.

Dynamical Systems: Lecture 1 19

Eigenvalues and Eigenvectors

Let A be an "×" square matrix mapping vectors from ℝ% to ℝ%.Eigenvalues and eigenvectors of A satisfy

'( = *(

• Eigenvalues l+ are found by solving the characteristic equation• Complex l+ are come in complex conjugate pairs• If real and distinct then there is a complete independent set of

eigenvectors (+ (one for each eigenvalue)

Dynamical Systems: Lecture 1 20

If there is a complete set of eigenvectors, then they span the vector space ℝ"This means that ANY vector # in ℝ" can be expressed as a weighted sum of eigenvectors:

# = %&'& + ⋯+ %"'"

If the eigenvalues are not distinct, there may not be a complete set of eigenvectors. (See Perko Chapter 1 on how to deal with this)

Dynamical Systems: Lecture 1 21

Eigenvalues and Eigenvectors

Matrix diagonalisa.on

• If a real matrix ! has " distinct real eigenvalues, then there is a complete set of real eigenvectors that span the vector space ℝ$

• The matrix ! is then directly diagonalizable!% = %' ⟹ ! = %'%)*

• ' is a diagonal matrix of eigenvalues, V is a matrix of eigenvectors.

Dynamical Systems: Lecture 1 22

Complex eigenvalues

• If a matrix has complex eigenvalues then its eigenvectors are complex, i.e. it cannot be diagonalized using matrices of real numbers

• For a 2×2 real matrix #:$ = & + (), $̅ = & − ()- = . + (/, 0- = . − (/

• Let V = [w, u], then123#1 = & −)

) &Dynamical Systems: Lecture 1 23

Example from the notes! = 3 −2

1 1

' = 2 + ), '̅ = 2 − ), = 1 + ) 1 -, ., = 1 − ) 1 -

3 −21 1 = 1 + ) 1 − )

1 12 + ) 00 2 − )

1 + ) 1 − )1 1

01

= 1 10 1

2 −11 2

1 10 1

01

This is a standard form for complex eigenvaluesDynamical Systems: Lecture 1 24

Linear Autonomous Systems

• A system of first order linear differential equations can be written in vector form

"̇ = $"Define

%$ ≝ '()*

+ $(,!

Then" . = %/$" 0

Dynamical Systems: Lecture 1 25

Compu&ng the matrix exponen&al

• Use eigenvalues! If the eigenvalues are real and distinct!" = $diag !)* $+,

If the eigenvalues are complex, use the previous expansion

!" = $!- +.. - $+, = $ !-cos2 −!-sin2

!-sin2 !-cos2 $+,

(see Perko ‘Differential equations and dynamical systems’ Sec.1.5)

Dynamical Systems: Lecture 1 26

Representing dynamical systems• Ordinary differential equations can be represented as ! coupled

first order differential equations

• Each of the ! unknowns is called a ‘state’, "# $ ∈ ℝ' $ ∈ ℝ( is the state vector

• "̇# = +# ",, ⋯ "( ,where each +# maps ' to a real number xi+#: ℝ( → ℝ

• If +# is defined on a subset of ℝ( (its domain), 7 ⊆ ℝ(, then +#: 7 → ℝ (e.g. " is only real for " ≥ 0, so 7 = {": " ≥ 0})

• f is the vector with =th element +=, i.e. f: ℝ( → ℝ( or f: 7 → ℝ(Dynamical Systems: Lecture 1 27

The set of non-linear differen1al equa1ons may now be wri8en as"̇ = $ "

This is an autonomous equa1on as it does not depend on 1me explicitly. The equa1on is linear if $ " = %" and % is an &×& real matrixNon-autonomous systems are of the form

"̇ = $ ", )and are not part of the course (there is one example in the first sheet as a warning of their difficulty)

Dynamical Systems: Lecture 1 28

Representing dynamical systems

Representing dynamical systemsParameters: ! may depend on a parameter vector " ∈ ℝ% where p does not necessarily equal &. For example, the equations may be those of motion dependent on a single mass and then ' = 1. We then write

+̇ = ! +; "

Maps or difference equations are not differential equations, but represent recurrence relations such as

+-./ = 0 +-; "These are written as

+ ↦ 0 +; "This is similar to the representation for register transfers in digital logic

Dynamical Systems: Lecture 1 29

A solu'on of a differen'al equa'on is a map from the 'me interval ! ∈ #, % to the space ℝ', passing through the ini'al condi'on ()at ! = 0:

(: #, % → ℝ' such that (̇ = / ((t); 4 and ( 0 = ()

Note that a < 0 and b > 0 if x0 is at ! = 0.

We will not solve such equa'ons – we will look at the geometry of the solu'ons.

Dynamical Systems: Lecture 1 30

Representing dynamical systems

Existence and uniqueness of solutions

• Does a solution exist? Is it unique?• The study of existence and uniqueness is highly technical – typically

part of a typical maths degree

• The lecture notes describe an aside (outside the course!)

considering existence (Lipschitz continuity) and dependence of

convergence on initial conditions and parameters (Gronwall’s

lemma) – see Perko sections 2.2 & 2.3

Dynamical Systems: Lecture 1 31