# DIFFERENTIAL EQUATIONS

Chapter 6. Bifurcation. In Section 1, we use examples, including Euler’s
buckling beam, to introduce the concept of bifurcation of critical points
of differential equations when some parameters are varied. In Section 2,
and node appear for this type of bifurcations. We analyze the geometric
aspects of some scalar differential equations that undergo saddle-node bifurcations
and use them to formulate and prove a result concerning saddle-node
bifurcations for scalar differential equations. In Section 3, we study transcritical
bifurcations and apply them to a solid-state laser in physics. Again,
the geometric aspects of some examples are analyzed and used to formulate
and prove a result concerning transcritical bifurcations for scalar differential
equations. In Section 4, we study pitchfork bifurcations and apply them
to Euler’s buckling beam and calculate Euler ’s first buckling load, which is
the value the buckling takes place. The hysteresis effect with applications
in physics is also discussed. A result concerning pitchfork bifurcations for
scalar differential equations is formulated using the geometric interpretation.
In Section 5, we analyze the situations where a pair of two conjugate complex
eigenvalues cross the pure imaginary axis when some parameters are
varied. We introduce the Poincar´e-Andronov-Hopf bifurcation theorem and
apply it to van der Pol’s oscillator in physics.

Chapter 7. Chaos
. In Section 1, we use examples, such as some discrete
maps and the Lorenz system, to introduce the concept of chaos. In Section
2, we study recursion relations, also called maps, and their bifurcation
properties by finding the similarities to the bifurcations of critical points of
differential equations, hence the results in Chapter 6 can be carried over.
In Section 3, we look at a phenomenon called period-doubling bifurcations
cascade, which provides a route to chaos. In Section 4, we introduce some
universality results concerning one-dimensional maps. In Section 5, we study
some properties of the Lorenz system and introduce the notion of strange
attractors. In Section 6, we study the Smale horseshoe which provides an
example of a strange invariant set possessing chaotic dynamics.

Chapter 8. Dynamical Systems. In Section 1, we discuss the need to
study the global properties concerning the geometrical relationship between
critical points, periodic orbits, and nonintersecting curves . In Section 2, we
study the dynamics in R2 and prove the Poincar´e-Bendixson theorem. In
Section 3, we use the Poincar´e-Bendixson theorem, together with other
results, to obtain existence and nonexistence of limit cycles, which in turn
help us determine the global properties of planar systems. In Section 4, we
apply the results to a Lotka-Volterra competition equation. In Section 5,
we study invariant manifolds and the Hartman-Grobman theorem, which
generalize certain results for planar equations in Chapter 4 to differential
equations in Rn.

Chapter 9. Stability. Part II. In Section 1, we prove a result concerning
the equivalence of “stability” (or “asymptotic stability”) and “uniform
stability” (or “uniform asymptotic stability”) for autonomous differential
equations. In Section 2, we use the results from Chapter 3 to derive stability
properties for general linear differential equations, and prove that they are
determined by the fundamental matrix solutions. The results here include
those derived in Chapter 5 for linear differential equations with constant or
periodic coefficients as special cases. Stability properties of general linear
differential equations with linear or nonlinear perturbations are also studied
using the variation of parameters formula and Gronwall’s inequality. In
Section 3, we introduce Liapunov’s method for general (nonautonomous) differential
equations and derive their stability properties, which extends the
study of stabilities in Chapter 5 for autonomous differential equations.

Chapter 10. Bounded Solutions. In Section 1, we make some definitions
and discuss the relationship between boundedness and ultimate boundedness.
In Section 2, we derive boundedness results for general linear differential
equations by using the results from Chapter 9. It will be seen that
stability and boundedness are almost equivalent for linear homogeneous differential
equations, and they are determined by the fundamental matrix
solutions. For nonlinear differential equations, examples will be given to
show that the concepts of stability and boundedness are not equivalent. In
Section 3, we look at the case when the coefficient matrix is a constant
matrix, and verify that the eigenvalues of the coefficient matrix determine
boundedness properties. In Section 4, the case of a periodic coefficient matrix
is treated. The Floquet theory from Chapter 3 is used to transform the
equation with a periodic coefficient matrix into an equation with a constant
coefficient matrix. Therefore, the results from Section 3 can be applied. In
Section 5, we use Liapunov’s method to study boundedness properties for
general nonlinear differential equations.

Chapter 11. Periodic Solutions
. In Section 1, we give some basic results
concerning the search of periodic solutions and indicate that it is appropriate
to use a fixed point approach. In Section 2, we derive the existence of
periodic solutions for general linear differential equations. First, we derive
periodic solutions using the eigenvalues of U(T, 0), where U(t, s) is the fundamental
matrix solution of linear homogeneous differential equations. Then
we derive periodic solutions from the bounded solutions. Periodic solutions
of linear differential equations with linear and nonlinear perturbations are
also given. In Section 3, we look at general nonlinear differential equations.
Since using eigenvalues is not applicable now, we extend the idea of deriving
periodic solutions using the boundedness. First, we present some Masseratype
results for one-dimensional and two-dimensional differential equations,
whose proofs are generally not extendible to higher dimensional cases. Then,
for general n-dimensional differential equations, we apply Horn’s fixed point
theorem to obtain fixed points, and hence periodic solutions, under the assumption
that the solutions are equi-ultimate bounded.

Chapter 12. Some New Types of Equations
. In this chapter, we
use applications, such as those in biology and physics, to introduce some
new types of differential equations, which are extensions and improvements
of the differential equations discussed in the previous chapters. They include
finite delay differential equations, infinite delay differential equations, integrodifferential
equations, impulsive differential equations, differential equations
with nonlocal conditions, impulsive differential equations with nonlocal
conditions, and abstract differential equations. For each new type of differential
equations mentioned above, we use one section to describe some of
their important features. For example, for integrodifferential equations, we
outline a method which can reformulate an integrodifferential equation as
a differential equation in a product space; and for abstract differential and
integrodifferential equations, we introduce the semigroup and resolvent operator
approaches. The purpose of this chapter is to provide some remarks
and references for the recent advancement in differential equations, which
will help readers to access the frontline research, so they may be able to
contribute their own findings in the research of differential equations and
other related areas.

How to use this book?

For an upper level undergraduate course. The material in Chapters
1–7 is enough. Moreover, if there are time constraints, then some results,
such as the following, can be mentioned without detailed proofs: in Chapter
2, the proofs concerning existence and existence without uniqueness of
solutions, the dependence on initial data and parameters, and the maximal
interval of existence; in Chapter 3, differential equations with periodic coefficients
and Floquet theory; in Chapter 5, the proofs concerning Liapunov’s
method; in Chapters 6–7, certain proofs concerning bifurcations and chaos.
(Note that Section 2.5 concerning the Fixed Point Method is optional.)

For a beginning graduate course. Chapters 1–11 provide a sufficient
resource for different selections of subjects to be covered. If time permits,
Chapter 12 can provide some direction for further reading and/or research
in the qualitative theory of differential equations.

One more thing we would like to point out is that Chapters 6 through 12
are rather independent of each other and the instructors may choose among
them to fit the last part of the course to their particular needs.

Exercises and notations. Most questions in the Exercises are quite important
and should be assigned to give the students a good understanding
of the subjects.

In Theorem x.y.z, x indicates the chapter number, y the section number,
and z the number of the result in section y. The same numbering system
holds true for Lemma x.y.z, Example x.y.z, etc.

Acknowledgments

First, I thank Professors ZuXiu Zheng and Ronald Grimmer for their inspirations
and for directing my Master thesis and Ph.D. dissertation, respectively,
in the area of Differential Equations. Then I thank my department
and college for supporting me during the planning and writing of the book.

This book grew from my class notes, so for their valuable comments I
thank my students: Kathleen Bellino, Paul Dostert, Roxana Karimianpour,
Robert Knapik, Justin Lacy, Florin Nedelciuc, Rebecca Wasyk, and Bruce
Whalen. My colleagues Carter Lyons and Esther Stenson read certain parts
of the first version and made some modifications which greatly improved the
exposition, so I thank them for their help. I also thank my colleague Bo
Zhang for informing me of some key reference books, and colleagues Carl
Droms, Jim Sochacki, and Paul Warne for helping with LaTex and Maple.

I sincerely thank the following reviewers of the manuscript:
Nguyen Cac, University of Iowa,
Przemo Kranz, University of Mississippi,
Jens Lorenz, University of New Mexico,
Martin Sambarino, University of Maryland,
Anonymous, Brigham Young University,
Anonymous, Lafayette College.
During the writing of the book, I was sometimes at “critical points”
(meaning “directionless” in ODEs). The comments and suggestions of the
reviewers gave me the direction I needed. Some reviewers pointed out errors
and confusing statements, and made specific recommendations to correct
them, which greatly improved the presentation of the book. However, I am
solely responsible for the remaining errors, if any, and invite the readers
to contact me with comments, recommendations and corrections, using the

I also thank George Lobell, Acquisitions Editor of Prentice Hall, for
accepting my humble first version and encouraging me to expand the first