Equilibrium Analysis for Systems

● Outline:● Solving linear systems● Classification of equilibria

Linear Systems

● In 1d phase space flows are extremely confined, 2d phase space allows for more diversity of behaviour

● Start with linear systems in 2d:

● x*=0 is always a fixed point● Solutions can be visualized as trajectories on

(x,y)-plane (“phase plane”)

x=ax+byy=cx+dy x=( xy) , ˙x=A x A=(a b

c d )or

Example: Harmonic Oscillator

● Point mass on a spring which is perturbed from equilibrium, no friction

● To derive the equation we use Newton's laws

F=ma=md2/dt 2 x

Linear Systems – Examples (1)

● Let's look through some examples to see what can happen

1.) “harmonic oscillator”

let ω2=k/m and v=dx/dt

md2/dt 2 x+kx=0

x = vv = −ω

2 x

x

v

closed orbits(~oscillations)

Linear Systems – Harmonic oscillator

1.) Could just solve system (->later)

2.) Solutions are on ellipses

Why?

x

v

● Why closed orbits?

ω2 x2

+v2=C

ddt

(ω2x

2+v

2 )=2ω2x x+2 v v

=2ω2 xv+2v (−ω2)

=0

Linear Systems – Examples (2)

2.)

● Essentially 2 decoupled equations so we know solutions

● Depending on a, how do the phase portraits look like?– Always have one “stable” direction -> y– Other direction may be stable or not

x = axy = − y

˙x=A x A=(a 00 −1)

x ( t) = x0 exp (at )y ( t) = y0 exp(−t )

Stable/Unstable Manifolds

● Directions which attract/repel flow into an equilibrium point

● DEF:● Stable manifold: ● Unstable manifold:

{x0: x( t)→x FP for t→∞}

{x0: x( t)→x FP for t→−∞}

Examples – Stable Nodes (a<0)

“slow” direction

“fast” direction

xy=x0/ y0 e

(a+1 )tsymmetrical node/

star

● Stable/Unstable manifolds?

Non-isolated Fixed points (a=0)

● Every point on real line is a FP (so there is no “gap” between FP's)

Saddle Points (a>0)

● One stable and one unstable direction

● Stable/Unstable manifolds?

Stability Language

● A fixed point (FP) x is● An attracting FP : all trajectories starting near x will

eventually approach it asymptotically ● A globally attracting FP: if all trajectories will

approach it asymptotically● A Liapunov stable FP: if all trajectories starting

close to x will remain close to x for all times● A neutrally stable FP: If x is Liapunov stable but

not attracting.● A stable FP: x is Liapunov stable and attracting● An unstable FP: x is not attracting

● Match examples from before?

Not Liapunov stable but attracting?

● Consider the following flow on a circle:

● All trajectories end up in FP, FP is attracting● There are trajectories that are arbitrarily close

to the FP, but don't remain close forever -> FP is not Liapunov stable

Θ=1−cos Θ ,Θ∈[0,2π ]

half stable FP

Classification of Linear Systems

● In previous examples x,y-directions played following roles:● Asymptotic directions of trajectories● “Straight-line” trajectories: an initial condition that

starts on them remains on them forever

● Analogue of straight-line trajectories?

● Put this into

x (t )=exp(λ t) v for some v≠0

˙x=A x

λ exp(λ t ) v=exp (λ t)A vλ v=A v

Eigenvectors/Eigenvalues (1)

● If straight-line solution exists v is an eigenvector for A and λ is the corresponding EV.

● How to find eigenvalues?● Solutions of the characteristic equation

● General solution in 2d:det (A−λ I )=0

Δ=det(A ) , τ=Tr (A)

λ1 /2=τ±√ τ2−4 Δ

2

Example Eigenvalues/vectors (1)

x = x+ yy = 4x−2y IC : (x0, y0)=(2,−3)

ddt (

xy)=(1 1

4 −2)(xy )

● Vector notation

det (1−λ 14 −2−λ)=0

1.) find char. eq.

(1−λ)(−2−λ)−4=0λ2+λ−6=0 2.) solve char. eq.

λ1 /2=−1 /2±√(25/ 4)

λ1=2, λ2=−3

Example Eigenvalues/vectors (2)

3.) Find eigenvectors,i.e. solve (1−λ1/2 1

4 −2−λ1 /2)( xy )=(0

0)

(−1 14 −4)( x

y )=(00) v 1=(1

1)

(4 14 1)(

xy )=(0

0) v 2=( 1−4 )

● General solution looks like:

x (t )=c1(11)exp (2t)+c 2( 1

−4 )exp(−3t)

Example Eigenvalues/vectors (3)

4.) Determine c1 and c

2

● Solution is:

( 2−3)=c1(11)+c2( 1

−4 )● Solve this linear system, e.g. (1)-(2) yields

5=5c2, i.e. c2=1

−3=c1−4, i.e. c1=1

x (t )=(11)exp(2t)+( 1−4)exp (−3t)

Example Eigenvalues/vectors (4)

● Phase portrait:

v 1

v 2(unstable mf)

(stable mf)

What if λ2<λ

1<0 ?

(λ1)

(λ2)

● Rules:● Trajectories tangent to slow direction for t->“+infinity”● Trajectories parallel to fast direction for t->“- infinity”

What if λ2=λ

1 ?

1.) λ=0 -> all plane filled with FP

2.) If there are two independent EV -> every vector is EV -> star node

3.) Only one EV (eigenspace is 1d) -> degenerate node

star node degenerate node

What if the EV's are complex?

● Must be complex conjugates● x,y are lin. comb.'s of

λ1 /2=α±iω

c1 /2 exp(α t) (cos (ω t)+ isin (ω t))

α<0 α=0 α>0

stable spiral center unstable spiral

Solving The Harmonic Oscillator (1)

● We started with

md2/dt 2 x+kx=0, say x (0)=0,dx /dt (0)=v0

x = vv = −ω

2 xwith ω2=k /m

A=( 0 1−ω

2 0) λ2+ω2=0→λ=±iω

Eigenvectors

(∓iω 1−ω

2∓iω)( xy)=0 (∓1/ω

1 )

Solving The Harmonic Oscillator (2)● Hence:

● Initial conditions:

● Which is a periodic motion along

(x ( t)v (t ))=c1(−i /ω

1 )exp (iω t)+c2( i /ω1 )exp(−iω t)

( 0v0

)=c1(−i /ω1 )+c2(i / ω1 ) c1=c2=v0/2

(x ( t)v (t ))=v0/2(−i /ω

1 )exp (iω t)+v0/2( i /ω1 )exp(−iω t)

(x ( t)v (t ))=(v0/ω sin (ω t)

v0 cos (ω t) ) e ix=cos (x )+isin (x )

(ω /v0)2 x2

+1 /v02 y2

=1

Solving The Harmonic Oscillator

● Some observations:● As seen the harmonic oscillator carries out a

periodic motion along an ellipse given by

(for x0=0 and dx/dt (0)=v0)● The trajectory depends on:

– The structure of the system (i.e. on ω)– ... but also on the initial conditions

● Hence: – trajectories of the harmonic oscillator are a family of

ellipses– If we perturb one trajectory a bit we don't return to the

same trajectory!

(ω /v0)2 x2

+1 /v02 y2

=1

Eventually ...General Classification

Δ=det(A ) , τ=Tr (A) λ1 /2=1/2 ( τ±√ τ2−4 Δ )

Summary

● What you should remember:● How to solve a 2(or higher)-dim linear system● How to draw phase portraits● Classify types of behaviour of 2d linear systems● “Speak the language of fixed points and stability”

(stable+unstable manifolds, attracting, globally attracting, Liapunov stable, stable, unstable, stable+unstable nodes and spirals, centers)

● In the seminar we will use this to understand the mathematics of love.