Introduction to Bifurcations and The Hopf Bifurcation Theorem for
Planar Systems
Roberto Muoz-Alicea
n
Department of Mathematics
Colorado State University
[email protected]
Report submitted to
Teetering Tail Rotor Instabilities
UNIVERSITY OF BRISTOL
DEPARTMENT OF ENGINEERING MATHEMATICS
TEETERING TAIL ROTOR INSTABILITIES
Daniel Segal (Engineering Mathematics)
Project thesis submitted in sup
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 li
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 i
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
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
E
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 equil
Lecture Notes on Nonlinear Vibrations
Richard H. Rand
Dept. Theoretical & Applied Mechanics Cornell University Ithaca NY 14853 [email protected]
http:/www.tam.cornell.edu/randdocs/
version 52
Copyright
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
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
C
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 plan
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
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 Lor
The Lorenz Equations
B.K. Muite
[email protected]
April 18, 2011
. References:
i. Doering and Gibbon,
Applied Analysis of the Navier-Stokes Equations,
Chapter 4
ii. Strogatz,
Nonlinear Dynamics and Chao
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
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 stud
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 El
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 SI
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,
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 vect
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
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 veloci
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
Attitude Control of Spacecraft
Attitude must be controlled to align instruments, antennae, solar panels, etc.
Therearethreebasicdevicestogetthisdone:
1.Attitudecontrolthrusters
2.Momentumwheels
image
Attitude Control of Spacecraft
Attitude must be controlled to align instruments, antennae, solar panels, etc.
Therearethreebasicdevicestogetthisdone:
1.Attitudecontrolthrusters
2.Momentumwheels
image
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
b
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