Dynamics and Bifurcations

In this opening chapter, we present selected basic concepts about the geometry of solutions of ordinary differential equations. To keep the ideas free from technical complications, the setting is one-dimensional—the scalar autonomous differential equations. Despite their simplicity, these concepts are central to our subject and reappear in various incarnations throughout the book. Following a collection of examples, we first state a theorem on the existence and uniqueness of solutions. Then we explain what a differential equation is geometrically. To facilitate qualitative analysis, geometric concepts such as vector field, orbit, equilibrium point, and limit set are included in this discussion. The next topic is the notion of stability of an equilibrium point and the role of linear approximation in determining stability. We conclude the chapter with an example of a scalar differential equation defined on a one-dimensional space other than the real line—a circle.

With the current proliferation of computers, numerical simulations have become common practice, suggesting new mathematical discoveries and new areas of applications. Despite the power of numerical approximation schemes as “experimental” tools and their case of implementation on the computer, there is always the difficulty of deciding on the accuracy of computations. Even in the case of a scalar differential equation, one can he confronted with rather strange mathematical phenomena. This is largely due to the fact that numerical approximation of a differential equation leads to a difference equation, and that difference equations, despite their innocuous appearance, can haw amazingly complicated dynamics. In this chapter, we first illustrate how difference equations, also called maps, arise in numerical approximations. Because of their importance in other contexts, we then undertake the study of dynamics and bifurcations of maps. In particular, we investigate local bifurcations of a class of maps, monotone maps, which will later play a prominent role in our study of differential equations. We end the chapter with a brief exposition of a landmark quadratic map, the logistic map.

In this chapter, we investigate the bifurcation behavior of periodic solutions of 1-periodic differential equations. We first point out that, in terms of Poincaré maps, the study of local bifurcations of periodic solutions is equivalent to the study of bifurcations of fixed points of monotone maps given in Section 3.3. Next, we develop selected ideas from the “method of averaging” and show how to compute higher- order derivatives of the Poincaré map about a fixed point. For practical purposes, we accomplish this solely in terms of the vector field and the 1-periodic solution. We then employ these results to determine local bifurcations of nonhyperbolic 1-periodic solutions of 1-periodic differential equations depending on a scalar parameter. We should warn you that this chapter, by necessity, is more technical than the previous ones.

With this chapter we commence our investigation of the geometry of planar autonomous differentia] equations. After pointing out how such equations arise in applications, we develop some necessary generalizations of certain geometric ideas which are reminiscent of the ones explored earlier for scalar equations. Because the simplest examples of planar systems are constructed by bundling a pair of scalar equations— product systems—we present a discussion of such systems, including the Flow Box Theorem. We also analyze the geometry of conservative systems as another class of vector fields with special properties. Finally, to give a hint of things to come, we present multiple examples of autonomous differential equations illustrating various bifurcations on the plane.

In this chapter, we undertake a detailed investigation of the rather special class of planar autonomous differential equations where the vector field is given by a linear map. By exploiting special properties of solutions of linear systems, with a small dose of linear algebra, we will be able to compute the flows of these systems explicitly and determine their phase portraits. After obtaining explicit solutions, we direct our attention to qualitative questions and classify linear systems up to flow equivalence. We also investigate certain bifurcation phenomena within the class of linear systems. We conclude the chapter with several useful facts about the solutions of nonautonomous linear differential equations. Admittedly, striving for explicit solutions may seem somewhat of a deviation from our earlier efforts. However, this is one of the few situations where such a complete answer, possessing its own mathematical appeal, exists. Furthermore, this information will be important in the local qualitative analysis of equilibrium points of nonlinear systems. The success of obtaining explicit solutions of linear systems is not, however, without an ironic disappointment. The important task of deciding the qualitative equivalence of two linear systems requires considerably more mathematics than mere formulas for their explicit solutions, as we shall see in this chapter.

In this chapter, we investigate the stability and instability properties of equilibrium points of planar differential equations. It is evident from our foregoing discussions that the stability type of an equilibrium point of a linear system is determined by the eigenvalues of its coefficient matrix. Analogous to the results in Section 1.3, we prove several theorems to show that, under certain conditions, the stability type of an equilibrium point of a nonlinear planar differential equation is determined by the linear approximation of the vector field in a sufficiently small neighborhood of the equilibrium point. In order to determine how large these “small” neighborhoods can be, we present another, somewhat more geometric, technique—the direct method of Liapunov—for investigating the stability of an equilibrium of a nonlinear system. We continue our presentation with an analysis of some of the finer geometric details of the flows of nonlinear systems in a neighborhood of an equilibrium point of saddle type. Next, we include a discussion of deciding the local equivalence of flows of nonlinear systems from that of their linear approximations. We conclude the chapter with an example illustrating the global dynamics of saddle points—saddle connections.

In this chapter, we undertake the study of stability and bifurcations of nonhyperbolic equilibria of a planar system in the case where the linearized vector field has one zero and one negative eigenvalue. Our investigation culminates in the observation that the local dynamics and bifurcations of such a planar system are determined from those of an appropriate scalar differential equation. Analysis of the resulting scalar equation can, of course, be accomplished using the results in Chapters 1 and 2. To provide a geometric view of this reduction from two dimensions to one, we include an exposition of a class of important invariant curves—center manifolds—which capture the asymptotic features of these planar systems.

In this chapter, we investigate the stability and bifurcations of a nonhyperbolic equilibrium point of a planar differential equation in the case where the linearized vector field has purely imaginary eigenvalues. Using polar coordinates, we capture the dynamics of such a system in the neighborhood of the equilibrium point in terms of the dynamics of an appropriate nonautonomous scalar differentia] equation with periodic coefficients. For the analysis of this scalar equation, we appeal to results in Chapters 4 and 5. When the vector field is subjected to small perturbations, the original equilibrium point persists, and there can be no new equilibria in the neighborhood. However, if the eigenvalues of the linearized system move away from the imaginary axis, one expects the equilibrium point to change its stability type. Thus change is typically marked by the appearance of a small periodic orbit encircling the equilibrium point. We present a proof of this celebrated result—the Poincaré-Andronov-Hopf Theorem—and a discussion of the stability of the periodic orbit. We conclude with an exposition of computational procedures for determining bifurcation diagrams of periodic orbits bifurcating from an equilibrium point.

After equilibrium points, the most interesting solutions of planar differential equations are periodic orbits. In fact, we have seen in the previous chapter the birth of periodic orbits when a nonhyperbolic equilibrium point undergoes a Poincaré-Andronov-Hopf bifurcation. There can also be periodic orbits far away from equilibrium points. The detection of such orbits is very difficult. In 1900, as part of problem sixteen of his famous list, Hilbert posed the following question: What is the number of (isolated) periodic orbits of a general polynomial system of differential equations on the plane? The problem remains unsolved even for the case where the components of the planar vector field are quadratic polynomials. Our inability to solve this basic problem exemplifies, in a striking way, the limited scope of our knowledge of periodic orbits. Despite the somber note, in this chapter we first present several basic theorems on the presence or absence of periodic orbits of planar systems. We then investigate the stability and local bifurcations of periodic orbits in terms of Poincaré maps. As an important application of these ideas, we establish the existence of a globally attracting periodic orbit of the oscillator of Van der Pol. We conclude the chapter with an example illustrating how a periodic orbit can bifurcate from a homoclinic loop.

In the numerous chapters that have come before, we have encountered many bifurcations, such as saddle-node for equilibria and periodic orbits, Poincaré-Andronov-Hopf, and breaking homoclinic loops and saddle connections. It is natural to ponder when, if ever, we will stop adding to the list and produce a complete catalog of all possible bifurcations. In this chapter, we indeed provide such a list for “generic” bifurcations of planar vector fields depending on one parameter. However, due to the overwhelming difficulty of the subject matter, our exposition, while precise, is devoid of verifications. To circumvent, certain technical complications, we confine our attention to a closed and bounded region of the plane, and in such a region characterize the structurally stable vector fields. To motivate this confinement, we then make a short digression to describe a class of vector fields whose dynamics are naturally confined to a bounded region—dissipative systems. Next, we explore the geometry of sets of mildly structurally unstable vector fields—first-order structural instability. By determining the sets of such vector fields forming hypersurfaces in the set of all vector fields, we arrive at a list of one-parameter “generic” bifurcations. You will undoubtedly notice that some of the familiar bifurcations are absent from the list. We provide an explanation for this as well, in terms of symmetries. We end the chapter with a glimpse into the intricate bifurcations of two-parameter vector fields.

In this chapter, we investigate the dynamics of two classes of vector fields with special characteristics—conservative and gradient. Both types of vector fields have the common property that they are defined in terms of functions; however, their flows are completely different. While periodic and homoclinic orbits may be omnipresent in conservative systems, the limit sets of orbits of gradient systems are necessarily part of the set of equilibria. We first uncover certain basic relations between the phase portraits of these systems and the geometry of underlying functions. Then we identify subsets of desirable “generic” functions. The vector fields of generic functions are structurally stable in the restricted sense that they are insensitive to small perturbations of the underlying functions. Analysis in the generic situations is made possible by the fact that the flows of both types of vector fields are essentially determined by the unstable manifolds of the saddle points. We also illustrate typical one-parameter bifurcations of conservative and gradient systems in nongeneric cases. Of course, the setting for the bifurcation theory of these systems has the important restriction that change of parameters preserve the conservative or gradient character of vector fields.

After about a dozen chapters on differential equations, we return here to the theme of Chapter 3 and explore, this time, some of the basic dynamics and bifurcations of planar maps. Our motives for delving into planar maps arc akin to the ones for studying scalar maps; namely, as numerical approximations of solutions of differential equations or as Poincaré maps. We begin our exposition with an introduction to the dynamics of linear planar maps. Then, following a section on linearization, we turn to numerical analysis and give examples of planar maps arising from “one-step” approximations of planar differential equations or from “two-step” approximations of scalar differential equations. Afterwards, we undertake, as usual, a detailed study of bifurcations of fixed points, including the Poincaré-Andronov-Hopf bifurcation for maps. The final part of the chapter is devoted to a synopsis of area-preserving maps, an important class arising from classical mechanics and possessing a rich history. The subject of planar maps is a vast one that is also mathematically rather sophisticated. Yet, many planar maps with innocuous appearances continue to defy satisfactory mathematical analysis. Indeed, the purpose of this modest, albeit long, chapter is to acquaint you with several famous planar maps and encourage you to explore their dynamics on the computer; for further mathematical nourishment, we will refer you to other sources.

In the final part of our book, we break the barrier of dimension two and venture into higher dimensions. Dynamical diversity in such dimensions is truly bewildering. Consequently, to bring our book to a conclusion in a finite number of pages, we attempt here to convey the current excitement of our subject with mere thumbnail sketches of several prominent examples. This chapter consists of abbreviated geometrical descriptions of two classical examples from the theory of forced oscillations: Van der Pol and Duffing. These nonautonomous planar systems contain a term that is a periodic function of time—hence the title of the chapter. Because of the time periodicity of the nonautonomous terms, the qualitative dynamics of these equations are studied most conveniently in the space IR² x S¹. Indeed, since in this space both of these equations possess global Poincaré maps, the results from the previous chapter become the natural mathematical backdrop. In the chaotic behavior of Duffing’s equation, the decisive role is played by transversal homoclinic points of its Poincaré map. We expound on this important connection by including a description of the dynamics of planar maps near such points.

In this chapter, we introduce four vector fields to illustrate selected highlights from three-dimensional dynamics and bifurcations. In the first example, a periodic orbit of a vector field in IR³ yields its stability to another periodic orbit of approximately twice the period. In the second example, as a periodic orbit becomes unstable, an invariant torus appears nearby. Using an appropriate Poincaré map, these two local bifurcations are, respectively, the counterparts of the period-doubling and Poincaré-Andronov Hopf bifurcations of a fixed point of a planar map from Chapter 15. The third example illustrates an important source of chaotic dynamics other than successive period-doubling bifurcations, a special type of homoclinic orbit—Sil’nikov orbit or saddle-focus. The final example features the Lorenz equations, the strange attractor mast, photographed.

This is the final chapter! At the same time, it is the beginning of a new geometric adventure into dimension four— the hyperspace. Arguably, the most natural differential equations residing in dimension four are the Hamiltonian systems with two degrees of freedom. Hence, we have chosen them as the subject of this chapter. Following a rapid introduction to the setting of Hamiltonian systems, we outline a topological program for the study of a small class of Hamiltonians—completely integrable systems—that can be analyzed successfully. From this contemporary viewpoint, we then study the flow of a pair of linear harmonic oscillators. Here, the term bifurcation gains yet another meaning in the context of level sets of the energy-momentum mapping. Our success with completely integrable systems is somewhat overshadowed by their rarity. Indeed, a satisfactory analysis of a general Hamiltonian system in four dimensions—unlike the case of the plane, one degree of freedom—is currently beyond reach. To hint at this complexity, we conclude the chapter with an example of a Hamiltonian that, in all likelihood, is nonintegrable.