Lectures on Periodic Orbits
11 February 2009
Most of the contents of these notes can found in any typical text on dynamical systems,
most notably Strogatz [1994], Perko [2001] and Verhulst [1996]. Complete proofs have been
omitted and wherever possible, r

Lecture 1B: Introdction to Perturbation Theory
CDS140b Lecturer: Wang Sang Koon
Winter, 2004
1
Basic Ideas
Perturbed and unperturbed problems: To nd approixmated solutions to a perturbed
problem, one will always assume that one have sucient knowledge of

Overview of CDS 140B:
Introduction to Dynamics
Wang Sang Koon
Control and Dynamical Systems, Caltech
koon@cds.caltech.edu
Textbook, Other Book and Papers
Textbook
Stephen Wiggins [2003]:
Introduction to Applied Nonlinear Dynamical Systems and
Chaos, Seco

CDS140B: Computation
of Halo Orbit
Wang Sang Koon
Control and Dynamical Systems, Caltech
koon@cds.caltech.edu
Importance of Halo Orbits: Genesis Discovery Mission
Genesis spacecraft will
collect solar wind from a L1 halo orbit for 2 1 years,
2
return th

Discussion on KAM Theorem
CDS140B Lecturer: Wang Sang Koon
winter, 2005
1
Integrable Systems
Involution.
Consider Hamiltonian equations
qi =
H
,
pi
with integrals F1 (q, p) and F2 (q, p) (where
cfw_F1 , F2 =
d
dt Fi
pi =
H
.
qi
= 0). The functions F1 an

1
CDS 140b: Homework Set 3
Due by Tuesday, February 10, 2009.
1. Structural stability and topological equivalence.
Consider the systems
y
x
x=
y=
and
x = y + x
y = x + y
for = 0.
(a) Let Ft , Gt : R2 R2 denote the ows dened by these two systems. Show that

Lecture 1B: Introdction to Perturbation Theory
CDS140b Lecturer: Wang Sang Koon
Winter, 2005
1
Basic Ideas
Perturbed and unperturbed problems: To nd approixmated solutions to a perturbed
problem, one will always assume that one have sucient knowledge of

Poincar-Lindstedt Method
e
CDS140B
Winter, 2005
1
Periodic Solutions of Autonomous 2nd Order Equations
Consider
x + x = f (x, x, ).
Assumptions:
periodic solutions exist for small, postive ;
requirements of the Poincar expansion thoerem have been satise

1
CDS 140b: Homework Set 1
Due by the end of the rst week after the rst unit
Problems.
1. Consider the following planar, autonomous vector eld:
x = x + y 2 ,
y = 2x2 + 2xy 2 ,
(x, y ) R2 .
(a) Prove that y = x2 is an invariant manifold for this vector eld

Introduction to Chaos and Symbol Dynamics
CDS140B Lecturer: Wang Sang Koon
winter, 2005
1
Introduction to Chaos.
The Study of Deterministic Chaos. Despite the fact that the system is deterministic, it has
the property that imprecise knowledge of the intia

Sigrid Leyendecker
Computational Dynamics for Mechanical Systems
1
Computational Dynamics for Mechanical Systems
two lectures of course CDS 140b: Introduction to Dynamics
Tuesday March 4 and Thursday March 6 2008, 10:30am-11:55am, Steele 214
Sigrid Leyend

Project Ideas: Lagrangian Coherent Structures
Philip du Toit
A. Prove the Flux Estimate Theorem:
1. Please refer to the paper: S. C. Shadden, F. Lekien, and J. E. Marsden, Denition
and properties of Lagrangian coherent structures from nite-time Lyapunov e

1
CDS 140b: Project Ideas (Hamiltonian Dynamics)
The following is a non-exhaustive list of possible project ideas for CDS-140b. The idea is
that you choose a topic during one of the guest lectures, and that you write a short summary
of it, demonstrating t

Normal Forms Theory
CDS140B Lecturer: Wang Sang Koon
Winter, 2004
1
Normal Form Theory
Introduction.
form.
To nd a coordinate system where the dynamical system take the simplest
The method is local in the sense that the coordinate transforms are generate

Dissipation-Induced Heteroclinic
Orbits in Tippe Tops
Nawaf Bou-Rabee
Jerrold E. Marsden
Louis A. Romero
12 February 2008
Abstract
This paper demontrates that the conditions for the existence of a
dissipation-induced heteroclinic orbit between the inverte

Poincar-Lindstedt Method
e
CDS140B
Winter, 2004
1
Periodic Solutions of Autonomous 2nd Order Equations
Consider
x + x = f (x, x, ).
Assumptions:
periodic solutions exist for small, postive ;
requirements of the Poincar expansion thoerem have been satise

1
CDS 140b: Homework Set 2
Due by Thursday, January 29, 2009.
1. Consider the system
x = y + x(r4 3r2 + 1)
y = x + y (r4 3r2 + 1)
where r2 = x2 + y 2 .
(a) Show that r < 0 on the circle r = 1 and r > 0 on r = 2. Use the Poincar
e
Bendixson theorem to show

1
CDS 140b: Homework Set 1
Due by Tuesday, January 20, 2009.
1. Determine whether the following system can have any periodic orbits:
x = y 2 + y cos x
and y = 2xy + sin x.
2. Use index theory to show that the system x = x(4 y x2 ), y = y (x 1) cannot have

Hopf Bifurcation
CDS140B Lecturer: Wang Sang Koon
Winter, 2005
1
Introduction
Consider
w = g (w, ).
where w Rn , Rp . Suppose it has a xed point at (w0 , 0 ), i.e., g (w0 , 0 ) = 0. Moreover, its
linearized equation
= Dw g (w0 , 0 )
has two purely imagin

Center Manifold Theory
CDS140B Lecturer: Wang Sang Koon
Winter, 2005
Introduction to Bifurcation Thoery.
In this chapter, we will cover the following materials:
Center Manifold Theory allows us to reduce the dimension of a problem, you will most
likely s

CDS 140b Final Project Introduction
Shane Ross, TA
February 2, 2004
This handout is meant to introduce you to the nal projects, which are due at the end of the term in
place of a nal exam. I will go over the steps in more detail as part of Thursdays lectu

The Poincar-Lindstedt Method:
e
the van der Pol oscillator
Joris Vankerschaver
jv@caltech.edu
The purpose of this document is to give a detailed overview of how the Poincar-Lindstedt
e
method can be used to approximate the limit cycle in the van der Pol s

Syllabus: Bifurcations
CDS-140b, 2009
Joris Vankerschaver
jv@caltech.edu
1
Structural Stability
Recall that a homeomorphism is a continuous bijection with a continuous inverse.
Denition 1.1. Let U be a domain in Rn . Consider vector elds X, Y X1 (U ) with

ALTEC
Dynamical S
H
C
C
Final Project
Shane D. Ross
Control and Dynamical Systems, Caltech
www.cds.caltech.edu/shane/cds140b
CDS 140b, February 5, 2004
Final Project
Issues to address in project
Equilibrium points
Periodic orbits
low order analytic

Bifurcation of Fixed Points
CDS140B Lecturer: Wang Sang Koon
Winter, 2005
1
Introduction
Consider
y = g (y, ).
where y Rn , Rp . Suppose it has a xed point at (y0 , 0 ), i.e., g (y0 , 0 ) = 0.
Two Questions: (1) Is the xed point stable or unstable? (2) Ho

The Method of Averaging
CDS140B Lecturer: Wang Sang Koon
Winter, 2003
1
Introduction
Remarks: The method leads generally to asymtotic series as opposed to convergent series. It is
not restriced to periodic solutions.
Averaging Method. Put the equation
x +

The Method of Averaging
CDS140B Lecturer: Wang Sang Koon
Winter, 2005
1
Introduction
Remarks: The method leads generally to asymtotic series as opposed to convergent series. It is
not restriced to periodic solutions.
Averaging Method. Put the equation
x +