Chapter 6Flows6.1 Definition of Dynamical SystemsDefinition: A dynamical system is a set of ordinar y differential equations ofthe formdxidt= Fi(x; c) (6.1)Here x = (x1, x2, ···, xn) ∈ Rnare called state variables and (c1, c2, ···, ck)∈ Rkare called control parameters. The functions Fi(x; c) are assumed to be‘sufficiently smoothe.’Remark: The state variables and control parameters are usually consideredto be in subspaces of euclidean spaces, but they may more generally be in n-and k-dimensional manifolds.Remark: The space of state variables is often called the phase space.Remark: Initial conditions for a ny solution of the dynamical system equa-tions belong in the phase space.Remark: The functions Fi(x; c) are usually assumed to be differentiableover the region of interest, but a Lipschitz condition |F (x; c) − F (x′; c)| <K(c)|x−x′| is sufficient for the most important properties of dynamical systemswhich we will explo it: the existence and uniqueness theorem (c.f., Sec. 6.2). Inmost cases of interest, the functions Fi(x; c) are polynomials in the xjwhosecoefficients depend on the control parameter s c. Such functions are Lipschitzon bounded domains in Rn.Remark: If the functions Fi(x; c) are not explicitly dependent on time (i.e.,∂Fi/∂t = 0, all i) the system is said to be autonomous. Otherwise, it is saidto be nonautonomous. It is often possible to replace a nonautonomous sys temof equations in n dimensions by an autonomous sy stem of equations in higherdimensions. We will see how this can be done in the two examples of per iodica llydriven systems discussed below.12 CHAPTER 6. FLOWS6.2 Existence and Uniqueness TheoremThe fundamental theorem for dynamical systems is the Ex istence and Unique-ness theorem, which we state here without proof.Theorem: If the dynamical system (6.1) is Lipschitz in the neighborhoodof a point x0, then there is a positive number s, andExistence: there is a function φ(t) = (φ1(t), φ2(t), ···, φn(t)) which satisfiesthe dynamical system equations dφi(t)/dt = Fi(φ(t); c) in the interval−s ≤ t ≤ +s, with φ(0) = x0;Uniqueness: the function φ(t) is unique.This is a local theorem. It guarantees that there is a unique nontrivial tra-jectory through each point at which a dynamical s ystem satisfies the Lipschitzproperty. If the dynamical system is Lipschitz throughout the domain of inter -est, this local theorem is easily extended to a global theo rem.Global Theorem: If the dynamical system (6.1) is Lipschitz, then there isa unique trajectory through every point, and that trajectory can be extendedasymptotically to t → ±∞.We cannot emphasize strongly enough how important the Existence andUniqueness theorem is for the topological discussion which forms the core ofthis work. Much of our discuss ion deals with closed periodic orbits in dynamicalsystems which ex hibit chaotic behavior. To be explicit, we depend heavily onthe fact that, in three dimensions, the topological organization of such orbitscannot change under any deformation. In or der for the organization to change,one orbit would have to pas s through ano ther. This means that at some stage ofthe deforma tion, two o rbits would pass through the same point in phase space.The exis tence and uniqueness theorem guarantees that this cannot occur. Moresp e c ifically, if two closed orbits share a single point, the two orbits must beidentical.6.3 Examples of Dynamical SystemsThe archetypical dynamical sys tem is the se t of equations representing the New-tonian motion of N independent point particles under their mutual interactions,gravitational or not. In three dimensions, this cons ists of n = 6N first ordercoupled ordinary differential equations: three for the coordinates and thre e forthe momenta of each particle. Under the gravitational interaction this systemis non Lipschitz: the gravitational potential energy diverges when any pair ofparticles becomes arbitrarily close.In the following four subsections we give four examples of dynamical systems.These are four of the most commonly studied dynamical systems. The examplesare pre sented in the historical order in which they were introduced.In the fifth subsection we give examples of some differential equations whichare not e quivalent to finite dimensional dynamical systems.6.3. EXAMPLES OF DYNAMICAL SYSTEMS 36.3.1 Duffing E quationThe Duffing equation was originally introduced to study the mechanical effectsof nonlinearities in nonideal springs. The force due to such a spring was assumedto be a perturbation of the general spring force law F (x) = −kx which retainedthe symmetry F (−x) = −F (x). The simplest perturbation of this form isF (x) = −kx − αx3with nonlinear spring ‘constant’ k + αx2. The spring is“harder” than an ideal spring if α > 0, “softer” if α < 0. The potential for thisnonideal spring force is even: U(x) =12kx2+14αx4. The equations of motionfor a mass m attached to a ma ssless nonideal spr ing, with damping constant γ,aredxdt= vmdvdt= F = −dUdx− γv (6.2)These two equations can be combined to a single second order equation in thevariable x:¨x + γ ˙x +1mdUdx= 0 (6.3)In what follows we set m = 1 for conve nience.The potential U(x) leads to exciting behavior if we relax the condition thatit represents a small perturbation of an ideal spring. That is, we assume thepotential may be unstable near the origin, but is globally stable. A family ofpotentials with these properties isA3: U(x) = −12λx2+14x4(6.4)Here the parameter λ defines the strength of the instability at the origin. Forλ > 0, the potential has an unstable fixed po int at x = 0 and two symmetricallydistributed stable fixed points at x = ±√λ. This potential describes manyphysical systems of interest: for example, a co lumn under sufficient stres s thatit buckles (c.f., Fig. 6.1). The dynamics of the system (6.3) are relativelysimple: Any initial condition eventually decays to one of the two ground states .However, if the oscillato r is periodically driven by including a forcing term ofthe form f cos ωt, the two coupled equations of motion becomedxdt= vdvdt= −dUdx− γ ˙x + f cos ωt (6.5)These two equations can be combined to a second or der equation in the singlevariable x:¨x + γ ˙x + x3− λx = f cos ωt (6.6)This equation for the driven damped Duffing oscillator leads to behavior whichis still not completely understo od.4 CHAPTER 6. FLOWSFigure 6.1: Top: A vertical beam under