CHAPTER 4 Periodic, Heteroclinic, and Homoclinic Orbits
I
n this chapter, we shift our attention away from equilibria, instead seeking more "interesting" solutions of nonlinear systems x = f (x). Much
CHAPTER 7 Introduction to Difference Equations
T
his Chapter concerns the dynamical behavior of systems in which time can be treated as a discrete quantity as opposed to a continuous one. For
example,
Guide to Commonly Used Notation
Symbol R Rn t u, v x, y x0 , y0 x , y f A, M, P D N tr( A) det( A) , Es , Eu , Ec Ws, Wu, Wc AB spancfw_v1 , v2 , . . . vn t (x0 ) t f : Rn Rm f Jf uv v 2 u-v 2 B(x, )
CHAPTER 13 The Laplace and Poisson Equations
U
p to now, we have dealt almost exclusively with pde s for which one independent variable corresponds to time. Now, we will analyze a pde
for which this i
CHAPTER 12 Introduction to Fourier Series
In each example, we were able to construct series representations of the solutions provided that the initial conditions themselves had special series represen
CHAPTER 11 Initial-Boundary Value Problems
T
he infinite spatial domains considered in the previous chapter give insight regarding the behavior of waves and diffusions. However, since such
domains are
CHAPTER 10 The Heat and Wave Equations on an Unbounded Domain
A
t first glance, the heat equation ut - u xx = 0 and the wave equation
to be positive constants, the only apparent distinction is between
CHAPTER 9 Linear, First-Order Partial Differential Equations
I
n this chapter, we will discuss the first of several special classes of pde s that can be solve via analytical techniques. In particular,
CHAPTER 8 Introduction to Partial Differential Equations
perature T depends not only upon time, but also upon spatial location. If x and y denote latitude and longitude and t denotes time, then the fu
CHAPTER 6 Introduction to Delay Differential Equations
I
n this Chapter, we turn our attention to delay differential equations (dde s), a major departure from the ordinary differential equations that
CHAPTER 5 Bifurcations
I
n practice, we often deal with ode s which contain parameters (unspecified constants) whose values can profoundly influence the dynamical behavior of
the system. For example,
CHAPTER 3 Nonlinear Systems: Local Theory
systems is usually impossible, so we must settle for qualitative descriptions of the dynamics. On the other hand, nonlinear systems can exhibit a wide variety
CHAPTER 2 Linear, Constant-Coefficient Systems
T
here are few classes of ode s for which exact, analytical solutions can be obtained by hand. However, for many systems which cannot be solved
explicitl
CHAPTER 1 Introduction
T
he mathematical sub-discipline of differential equations and dynamical systems is foundational in the study of applied mathematics. Differential equations
arise in a variety o
Ordinary and Partial Differential Equations
An Introduction to Dynamical Systems
John W. Cain, Ph.D. and Angela M. Reynolds, Ph.D.
Mathematics Textbook Series. Editor: Lon Mitchell 1. Book of Proof by