# 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,

we study saddle-node bifurcations and use examples to explain why saddle

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 R

^{2}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 R

^{n}.

**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.

**. In this chapter, we**

Chapter 12. Some New Types of Equations

Chapter 12. Some New Types of Equations

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,

Michael Kirby, Colorado State University,

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

address or email given below.

I also thank George Lobell, Acquisitions Editor of Prentice Hall, for

accepting my humble first version and encouraging me to expand the first

version to include additional subjects.

The production phase has been more involved than I anticipated. However,

with the help of my production editor Bob Walters, copyeditor Elaine

Swillinger, LaTex expert Adam Lewenberg, and Adobe software expert

Bayani DeLeon of Prentice Hall, it went smoothly. I thank them for giving

detailed instructions and for doing an excellent job in helping to make this

book a reality .

Finally, I thank my wife Tina and daughter Linda for their understanding,

support, and help.

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