equilibrium analysis for systems
TRANSCRIPT
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.