Introduction to Bifurcations and The Hopf Bifurcation Theorem for
Planar Systems
Roberto Muoz-Alicea
n
Department of Mathematics
Colorado State University
munoz@math.colostate.edu
Report submitted to Prof. I. Oprea for Math 640, Fall 2011
Abstract. We int
Teetering Tail Rotor Instabilities
UNIVERSITY OF BRISTOL
DEPARTMENT OF ENGINEERING MATHEMATICS
TEETERING TAIL ROTOR INSTABILITIES
Daniel Segal (Engineering Mathematics)
Project thesis submitted in support of the degree of Master of Engineering
Supervisors
Stability Analysis of Hills equation
Svetlana V. Simakhina
July 20, 2003
Acknowledgments
I would like to thank my advisor Charles Tier for all his valuable help, suggestions
and directions. I would like to express my gratitude to Victor Tarakanov (Russia)
ME 406 The Lorenz Equations
sysid
Mathematica 4.1.5, DynPac 10.67, 482002
intreset; plotreset;
1. Introduction
This notebook contains all of the material given in class on the Lorenz equations, and it constitutes section 2.5 of the class notes. The Loren
Chapter 2
Review of Lyapunov Functions
Abstract We turn next to some of the basic notions of Lyapunov functions.
Roughly speaking, a Lyapunov function for a given nonlinear system is a
positive denite function whose decay along the trajectories of the sys
Linear Dynamics, Lecture 1
Review of Hamiltonian Mechanics
Andy Wolski
University of Liverpool, and the Cockcroft Institute, Daresbury, UK.
November, 2012
Introduction
Joseph John Thomson, 1856-1940
Early accelerators were fairly straightforward.
Linear D
CDS140a
Nonlinear Systems: Local Theory
02/01/2011
3
Stability and Lyapunov Functions
3.1
Lyapunov Stability
Denition:
x0 of (1) is stable
t (x) N (x0 ).
An equilibrium point
and
t 0,
we have
An equilibrium point
> 0,
if for all
An equilibrium point
that
Lecture Notes on Nonlinear Vibrations
Richard H. Rand
Dept. Theoretical & Applied Mechanics Cornell University Ithaca NY 14853 rhr2@cornell.edu
http:/www.tam.cornell.edu/randdocs/
version 52
Copyright 2005 by Richard H. Rand
1
R.Rand
Nonlinear Vibrations
Proceedings of the International Conference on Theory and Applications of
Mathematics and Informatics - ICTAMI 2004, Thessaloniki, Greece
ON SOME PROPERTIES OF THE SYMPLECTIC AND
HAMILTONIAN MATRICES
by
Dorin Wainberg
Abstract: In the first part of the pa
Marshall University
Marshall Digital Scholar
Theses, Dissertations and Capstones
1-1-2011
Numerical Calculation of Lyapunov Exponents for
Three-Dimensional Systems of Ordinary
Differential Equations
Clyde-Emmanuel Estorninho Meador
cemeador@gmail.com
Foll
Hopf Bifurcations in 2D
2 ways for stable xed point to lose stability:
1 real eigenvalue passes through = 0
(zero-eigenvalue bifurcations),
2 complex conjugate eigenvalues cross into right half plane
(Hopf bifurcations).
Supercritical Hopf bifurcation o
THREE DIMENSIONAL
SYSTEMS
Lecture 6: The Lorenz
Equations
6. The Lorenz (1963) Equations
The Lorenz equations were originally derived
by Saltzman (1962) as a minimalist model
of thermal convection in a box
x = (y x)
(1)
y = rx y xz
(2)
z = xy bz
(3)
where
Introduction to Lorenz's System of Equations
Nicholas Record
December 2003, Math 6100
1 Preliminaries
1.1 Initial Conditions
This paper discusses some fundamental and interesting properties of the Lorenz
equations, a topic which is well outside the scope
The Lorenz Equations
B.K. Muite
muite@umich.edu
April 18, 2011
. References:
i. Doering and Gibbon,
Applied Analysis of the Navier-Stokes Equations,
Chapter 4
ii. Strogatz,
Nonlinear Dynamics and Chaos,
Chapter 9
iii. Alligood, Sauer and Yorke,
Chaos: An
Harmonic Balance, Melnikov Method and
Nonlinear Oscillators Under Resonant Perturbation
Michele Bonnin1
1
Politecnico di Torino, Torino, Italy
SUMMARY
The Subharmonic Melnikovs method is a classical tool for the analysis of subharmonic orbits
in weakly p
Lagrangian and Hamiltonian Mechanics
D.G. Simpson, Ph.D.
Department of Physical Sciences and Engineering
Prince Georges Community College
December 5, 2007
Introduction
In this course we have been studying classical mechanics as formulated by Sir Isaac New
Nonlinear Dynamics 23: 6786, 2000.
2000 Kluwer Academic Publishers. Printed in the Netherlands.
Homoclinic Connections in Strongly Self-Excited Nonlinear
Oscillators: The Melnikov Function and the Elliptic
LindstedtPoincar Method
MOHAMED BELHAQ1, BERNOLD
PRAMANA
_journal of
physics
Printed in India
Vol. 45, No. 2,
August 1995
pp. 149-164
Time dependent canonical perturbation theory II: Application to the
Henon-Heiles system
M ITAXI P MEHTA and B R SITARAM
Physical Research Laboratory, Navrangpura, Ahmeda
Outline
0.3
Stability Denitions
g
RW
0.2
0.1
Lyapunov Functions
0.0
-0.1
-0.2
Velocity-based feedback
An
20
-0.3
Position-based feedback
0
200
300
500
tim
400
40
100
University of Colorado at Boulder
ba
Gim
Linear Closed-Loop Dynamics
eg]
s [d
ngle
l
ME 510 Homework (graduate students)
Hmk #1 (due 1/31): S&J: 1.12, 1.14, 1.15 (for these write result in the simplest possible form using any
coordinate frame or combination of coordinate frames), 3.1, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10
Hmk #2 (due 2/11): Homew
Statistical Attitude
Determination
Wahbas Problem
Assume we have N>1 observation measurements (i.e. measured
directions to sun, magnetic eld, stars, etc.), and we know the
corresponding inertial vector directions. Then we can write attitude
determination
Outline
Vector Notation
Vector Differentiation
B
Lots of brushing up on this material on your
own!
n
r si
O
rP
A
Colorado
University of Colorado at Boulder
Aerospace Engineering Sciences Department
10
Vector Notation
Hopefully a boring topic for you by
Stability of Rigid Body Rotations
Let the body axes (with unit vectors , , ) coincide with the principal axes with principal moments of
inertia , , (all distinct). Also, let the nominal angular velocity be
Assume a small perturbation is applied such that
Principal Rotation Vector
The building block of many advanced attitude coordinates.
Theorem 3.1 (Eulers Principal Rotation): A rigid
body or coordinate reference frame can be brought
from an arbitrary initial orientation to an arbitrary nal
orientation by
Attitude Control of Spacecraft
Attitude must be controlled to align instruments, antennae, solar panels, etc.
Therearethreebasicdevicestogetthisdone:
1.Attitudecontrolthrusters
2.Momentumwheels
image courtesy of
image courtesy of
http:/ipp.nasa.gov/innova
Attitude Control of Spacecraft
Attitude must be controlled to align instruments, antennae, solar panels, etc.
Therearethreebasicdevicestogetthisdone:
1.Attitudecontrolthrusters
2.Momentumwheels
image courtesy of
image courtesy of
http:/ipp.nasa.gov/innova
Introduction
Attitude coordinates are set of coordinates that
describe of both a rigid body or a reference frame
An innite number of coordinate choices
exists, same as with position coordinates
b3
b2
b1
A good choice in attitude coordinates can
greatly
Deterministic Attitude
Determination
Introduction
Attitude determination is broken up into two areas
Static attitude determination: All measurements are taken at the same time. Using
this snap shot in time concept, the problem becomes up of optimally so