Second order systems

Multiple scale technique

Second order systems

Lesson 11

Weakly nonlinear oscillator

d²u

dt² + ω0²u = εF (u, du dt ).

it is a general form of Duffing’s equation

d²u

dt² + u ∓ εu³ = 0 with ^F _{= ±u}³ or van der Pol’s oscillator

d²u

dt² + u − ε(1 − u²)du

dt = 0 with ^F _{= (1 − u}^2)du

dt .

For the first equation the method of renormalization

Type of solutions of the second order ODE.

The multiple scale technique permits successfully treat all types of weakly nonlinear oscillators. But

before introducing the method we review the types of the the second order ODE.

We start from the unperturbed equation

d²u

dt² + ω0²u = 0.

Its solution is

1 du

The phase plane

Let us introduce the coordinate system with ^{x = u} and

y = ^du_dt . Then

x = u = A cos ω⁰t + B sin ω⁰t

and

y = du

dt = −ω⁰A sin ω⁰t + ω⁰B cos ω⁰t.

Eliminating ^t yields the solution curve on ^{(x, y)} plane

x² + y²

ω₀² = A² + B².

The phase plane

The curves ^{x2 + y2 = A2 + B2} are ellipses with

The phase plane

The differential equation ^d_dt^{2u2 + du}_dt ^{+ 2u = 0} has the general solution

u = e^−2t 

A cos

√7

2 t + B sin

√7 2 t

where

A = u(0) and ₋ ^A

2 +

√7B

2 = du dt (0)

The phase plane

as we see the solutions are the spiral into the origin

Linear autonomous equations

There are three types of behavior of solutions of autonomous linear differential equations:

1. periodic orbits,

2. decay towards the origin, 3. diverge to the infinity.

Nonlinear autonomous equations

The nonlinear autonomous differential equations has one more special type of solutions limit cycle.

Occurrence of this type depends both on the form of the equation and on the initial conditions. As an

example consider the equation

d²u

dt² = −u + 

1 − u² − du dt

²du dt .

Nonlinear autonomous equations

Let us write the equation ^d_dt^2u2 _{= −u +} ^_{1 − u}² ₋ ^du_dt
^2du_dt using the phase plane variables

x = u, y = dx

dt = du dt .

Then the equation is equivalent to the system

dx

dt = y | · x

dy

dt = −x + (1 − x² − y²)y | · y

Nonlinear autonomous equations

d

dt(x² + y²) = 2(1 − x² − y²)y².

Using the polar coordinates

x = r cos θ r = sin θ

we get the

d

dt(r²) = 2(1 − r²)r² sin² θ.

Nonlinear autonomous equations

There is three possibility for the equation

d

dt(r²) = 2(1 − r²)r² sin² θ

1. If ^{r = 1} then _dt^{d (r2) = 0} and ^{r = const} give the periodic solution.

2. If ^{r > 1} then _dt^{d (r2) < 0} and the solution decrease to the periodic solution.

3. If ^{r < 1} then _dt^{d (r2) > 0} and the solution increase to the periodic solution.

This situation called the stable limit cycle.

Limit cycle

Types of solutions for nonlinear ODE

The solutions of nonlinear autonomous second order differential equations belong to the following four

classes

1. periodic solutions, 2. limit cycles,

3. solutions tends towards a fixed value 4. solutions tends to infinity.

The renormalization is success only for the first type of solutions.

Limitation of renormalization

Let us compare two problems

d²u

dt² + u = εu, u(0) = 1, du

dt (0) = 0 d²v

dt² + v = −εdv

dt , v(0) = 1, dv

dt (0) = 0

We look for the two term expansions

u(t, ε) = u⁰(t) + εu¹(t)

and

Limitation of renormalization

The second coefficients are following

d²u¹

dt² + u¹ = cos t, u1(0) = du¹

dt (0) = 0 ⇒ u1 = t

2 sin t, d²v¹

dt² + v¹ = − sin t, v¹(0) = dv¹

dt (0) = 0 ⇒ v¹ = 1

2(−t cos t + sin t).

Limitation of renormalization

Introducing the strained coordinates ^{t = s + εf1}, we get for the first equation

cos t + 1

2εt sin t = cos(s + εf¹) + 1

2εs sin s

= cos s − εf¹ sin s + 1

2εs sin s

Choosing ^{f1 =} 2^s we have ^u = cos s + O(ε) where

t = s + εs

2 + O(ε²)

Limitation of renormalization

Expressing ^s

s = t

1 + ε

2 + O(ε²)⁻¹

= t

1 − ε

2 + O(ε²) .

Thus

u = cos t

1 − ε

2 + . . .  .

The exact solution is

u = cos t√

1 − ε = cos t

1 − ε

2 + . . .  .

The renormalization provides a means of evaluating

Limitation of renormalization

Introducing the strained coordinates ^{t = s + εf1} for the second equation

cos t − 1

2ε(t cos t − sin t) = cos(s + εf¹) − 1

2ε(s cos s − sin s)

= cos s − εf¹ sin s − 1

2εs cos s + 1

2ε sin s

f¹ = −^s2 cot s is not acceptable since it ia singular for

s = π, 2π, 3π, . . .. Trying to make the expansion of solution uniform the strained expansion becomes nonuniform. Let us compare with the exact solution.

Time changing amplitude

u = e^−εt2 

cos r

1 − ε²

4 t + ε 2

q

1 − ^ε42

sin r

1 − ε² 4 t

 .

It is not a periodic solution since the frequency of oscillation and the amplitude are changed.

The frequency of oscillation changed from ¹ to ^q_{1 −} ^ε4²

The amplitude ^e−εt2 decays.

Let us see why the renormalization failed in general case.

Limitation of renormalization

d²u

dt² + u = εF (u, du

dt ), u(0) = a, du

dt (0) = 0

Starting from two term expansion ^{u0 + εu1} we get

u⁰ = a cos t and

d²u¹

dt² + u¹ = εF (a cos t, −a sin t).

Since ^F is a periodic function, it can be decomposed into the Fourier series

Limitation of renormalization

The homogeneous solution is ^A cos t + B sin t, so the term ^{a1 cos t + b1 sin t} will generate a particular solution

t

2(a¹ sin t − b¹ cos t).

The two term expansion is

a cos t + ε t

2(a¹ sin t − b¹ cos t) + nonsecular terms^ Using the renormalization ^{t = s + εf1} we get

s nonsecular terms

Limitation of renormalization

We require ^{f1 = a}2¹a^s − b¹s cot ₂^s_a. We see that iff ^{b1 = 0} we avoid the problems with ^cot 2^sa. In this case the expansion is

u = a cos s + O(ε), t = s + εa¹s

2a + O(ε²) ⇒ u = a cos t

1 − εa¹

2a + . . .  .

The expansion can generate only periodic solutions.

Limitation of renormalization

For Duffing’s equation we had ^F _{= −u}³, so

F(a cos t, −a sin t) = −a³ cos³ t = a³

4 (3 cos t + cos 3t)

and renormalization gave the positive result.

For van der Pol’s oscillator ^F _{= (1 − u}^2)du_dt

F = (1 − a² cos² t)(−a sin t) = (−a + a³

4 ) sin t + a³

4 sin 3t.

If ^a _{6= 2} the renormalization fails. The case ^{a = 2}

corresponds to the limit cycle and the normalization is

The end