Perturbation Theory &Stability Analysis
T. Weinhart, A. Singh, A.R. ThorntonMay 17, 2010
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1 Perturbation Theory
2 Algebraic equationsRegular PerturbationsSingular Perturbations
3 Ordinary differential equationsRegular PerturbationsSingular PerturbationsSingular in the domain
4 The non-linear springNon-uniform solutionUniform solution using Linstead’s MethodPhase-space diagram
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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Introduction
Perturbation TheoryConsiders the effect of small disturbances in the equation to thesolution of the equation.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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A first exampleLet x− 2 = 0. Then x = 2. What about
x− 2 = ε cosh(x)?
Assume ε 1 and substitute x = x0 + εx1 + ε2x2 + . . .
(x0 + εx1 + . . . )− 2 = ε cosh(x0 + εx1 + . . . )
and solve separately for each order of ε:
. . . ⇒ x ≈ 2 + ε cosh(2)
ConclusionsThe effect of the small parameter ε is small; therefore theperturbation is said to be regular. Otherwise we call it singular.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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Order symbol notation
Let f(x), g(x) be two functions defined around x = x0.We say, f = o(g) as x→ x0 iff lim
x→0f/g = 0.
We say, f = O(g) (of order) as x→ x0 iff limx→0
f/g = const 6= 0.
We say, f ∼ g (goes like) as x→ x0 iff limx→0
f/g = 1.
Examples:1 x2 = o(x) as x→ 0, b/c x2/x→ 0.2 3x2 = O(5x2) as x→ 0, b/c (3x2)/(5x2)→ 3/5.3 sin(x) ∼ x as x→ 0, b/c sin(x)/x→ 1.4 sin(x) = x+O(x3) as x→ 0, b/c (sin(x)− x)/x3 → 1/6.
5 sin(x) ∼ x− x3
3! as x→ 0, b/c sin(x)/(x− x3/3!)→ 1.So the solution to x− 2 = ε cosh(x) is x ∼ 2 + ε cosh(2) as ε→ 0.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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The Fundamental Theorem of Perturbation TheoryIf
A0 +A1ε+ · · ·+Anεn +O(εn+1) = 0
for ε→ 0 and A0, A1, . . . independent of ε, then
A0 = A1 = · · · = An = 0.
That is why we could solve separately for each order of ε:
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Regular Perturbations 6/21
Regular Perturbations
x2 − 3x+ 2 + ε = 0, ε 1.
Assume the roots have expansion x = x0 + εx1 + ε2x2 + . . . :
(x0 + εx1 + . . . )2 − 3(x0 + εx1 + . . . ) + 2 + ε = 0,
. . . ⇒ x = 1 + ε+ ε2 +O(ε3), 2− ε− ε2 +O(ε3).
Compare with exact solution,
x =3±√
1− 4ε2
=3± (1− 2ε− 2ε2 + . . . )
2
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Singular Perturbations 7/21
Singular Perturbations
εx2 − 2x+ 1 = 0, ε 1.
Assume the roots have expansion x = x0 + εx1 + ε2x2 + . . . :
ε(x0 + εx1 + . . . )2 − 2(x0 + εx1 + . . . ) + 1 = 0.
. . . ⇒ x = 1/2 + 1/8ε+ 1/16ε2 +O(ε3)
But where is the second solution? Compare with exact solution,
x =1±√
1− εε
=1± (1− ε/2− ε2/8 + . . . )
ε
. . . ⇒ x = 1/2 + 1/8ε+O(ε2), 2/ε− 1/2 +O(ε).
We can find the second solution by solving for w = εx, so
w2 − 2w + ε = 0.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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Ordinary differential equations
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Regular Perturbations 9/21
Regular Perturbations
x+ x = εx2, x(0) = 1.
Assume the roots have expansion x = x0 + εx1 + ε2x2 + . . . :
(x0 + εx1 + . . . ) + (x0 + εx1 + . . . ) = ε(x0 + εx1 + . . . )2,x0(0) + εx1(0) + · · · = 1.
. . . ⇒ x = e−t + ε(e−t − e−2t) +O(ε2).
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Singular Perturbations 10/21
Singular PerturbationsHeat equation: Temperature x for a stone in water
εx = 1− x, x(0) = 0.
Assume the roots have expansion x = x0 + εx1 + ε2x2 + . . . :
(x0 + εx1 + . . . ) = 1− (x0 + εx1 + . . . ).
. . . ⇒ x0 = 1 (Does not satisfy initial conditions).
Again, We can find a solution by transforming, τ = ε−1t,
d
dτx = 1− x, x(0) = 0.
. . . ⇒ x = 1− e−2τ = 1− e−2t/ε.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Singular in the domain 11/21
Singular in the domain
x+ εx2 = 1, x(0) = 0.
Assume the roots have expansion x = x0 + εx1 + ε2x2 + . . . :
(x0 + εx1 + . . . ) + ε(x0 + εx1 + . . . )2 = 1.
. . . ⇒ x ∼ t− εt3
3− ε2 2t5
15.
This solution is non-uniform (blows up for large t).x ∼ t− ε t33 is an asymptotic expansion for t <
√3/ε.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
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The non-linear spring
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Non-uniform solution 13/21
Duffing’s equationConsider a springs with a small non-linear perturbation
x+ ω2x+ εx3 = 0, x(0) = a, x(0) = 0.
with ω2 = k/m12 and ε 1. Expand:
x0 + ω2x0 = 0 ⇒ x0 = a cos(ωt) (1)
x1 + ω2x1 = −a3
4( 3 cos(ωt)︸ ︷︷ ︸secular term
+ cos(3ωt)︸ ︷︷ ︸no problem
). (2)
. . . ⇒ x ∼ a cos(ωt) + εa3
32ω2(cos(3ωt)− cos(ωt)− 12tω sin(ωt)).
Again, this blows up for large t. How can we find a uniformsolution?
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Uniform solution using Linstead’s Method 14/21
Linstead’s Method1 Assume the system has a natural frequency Ω.2 Transform to a strained time variable τ = Ωt.3 Expand solution x = x0 + εx1 + . . . as well as frequency
Ω = Ω0 + εΩ1 + . . . .4 Use the new variables Ωn to remove secular terms.
For Duffing’s equation we obtain
x = a cos(τ) + εa3
36ω2(cos(3τ)− cos(τ)) + . . .
with Ω = ω + 3a2
8ω ε+ . . . .
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
Phase-space diagram 15/21
Phase-space diagram
Any autonomous (no explicit t-dependence) second-ordersystem
x = f(x, x).
can be written as
x = y,
y = f(x, y)
Thendy
dx=y
x=f(x, y)y
.
A plot of in the xy plane is called a phase-space diagram.Solution plots are called orbits.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
For the linear spring, x+ω2x = 0, we obtain 12y
2 = −12ω
2x2 + c.
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
Figure: Phase space diagram for the linear spring with ω = 1.
Phase-space diagram 17/21
Phase-space diagram
Properties of phase-space diagrams• A closed orbit corresponds to a periodic system• Points where x = y = 0 are called critical points and
represent equilibrium points of the system.• Orbits cannot intersect or cross• This plot shows only the orbit structure, all temporal
information is lost.
Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring
For the linear spring with dissipation, x+ 2γx+ ω20x = 0, we
obtain x = e−γt (a cos(ωt) + b sin(ωt)) , ω2 = ω20 − γ2.
-1.0 -0.5 0.5 1.0
-1.0
-0.5
0.5
1.0
Figure: Phase space diagram for the linear damped spring withω0 = 2γ = 1.
For the Hertian spring, x+ ω2|x|3/2 = 0, we obtain12y
2 = −25ω
2|x|5/2 + c.
-1.0 -0.5 0.0 0.5 1.0
-1.0
-0.5
0.0
0.5
1.0
Figure: Phase space diagram for the Hertzian spring with ω = 1.
For Duffings equation, x+ ω2x+ εx3 = 0, we obtain12y
2 = −12ω
2x2 − 14εx
4 + c.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Figure: Phase space diagram for the non-linear spring with ω = ε = 1.
Phase-space diagram 21/21
Classification of Equilibrium PointsWe classify equilibrium points by linearizing the system aroundthem. We obtain the linear system
x = ax+ by,
y = cx+ dy.
Writing this as a second order system, we get
x− (a− d)x+ (ad− bc)x = 0
x = aeλt are solutions to this equation if λ satisfies
λ2 − (a− d)λ+ (ad− bc) = 0
Thus if solutions λ1, λ2 are distinct,
x = a1eλ1t + a2e
λ2t
We distinguish six types of critical points: [blackboard]Perturbation Theory Algebraic equations Ordinary differential equations The non-linear spring