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Ordinary &amp; Partial Differential Equations - Reynolds (2000) - Chapter 4 - Periodic, Heteroclinic

# Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 4 - Periodic, Heteroclinic

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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 of our discussion involves planar systems (i.e., f : R 2 R 2 ), because such systems admit particularly simple criteria for the existence of periodic solutions. 4.1. Periodic Orbits and the Poincaré-Bendixon Theorem A non-equilibrium solution x of the system x = f ( x ) is periodic if there exists a positive constant p such that x ( t + p ) = x ( t ) for all time t . The least such p is called the period of the solution, and tells us how often the solution trajectory “repeats itself”. In the phase portrait for the system of ode s , periodic solutions (sometimes called periodic orbits ) always appear as closed curves. On the other hand, not every closed curve corresponds to a periodic solution, as we shall see when we discuss homoclinic orbits. For linear, constant-coefficient systems, we learned to associate pure imaginary eigenvalues with periodic solutions. Determining whether a nonlinear system has periodic solutions is less straightforward. In preparation for stating criteria for existence of periodic solutions, we review some basic notions from calculus. Suppose that Γ ( t ) = ( γ 1 ( t ) , γ 2 ( t )) is a parametrized curve in R 2 ; i.e., x = γ 1 ( t ) and y = γ 2 ( t ) . Assume that γ 1 ( t ) and γ 2 ( t ) are continuously differentiable. At a given time t = t 0 , the tangent vector to the curve Γ ( t ) is given by Γ ( t 0 ) = ( γ 1 ( t 0 ) , γ 2 ( t 0 )) . 122

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periodic , heteroclinic , and homoclinic orbits 123 Γ (t) (t ) o Γ Γ (t) n τ inward outward (a) (b) Figure 4 . 1 . (a) Illustration of a tangent vector τ and normal vector n to a curve Γ at a particular time t = t 0 . (b) For the closed curves that we shall consider, each point on the curve has exactly one outward unit normal vector and one inward unit normal vector. Definition 4 . 1 . 1 . Any non-zero vector n ( t 0 ) in R 2 which is perpendicular to the tangent vector Γ ( t 0 ) is called a normal vector to the curve at t = t 0 . If n 2 = 1, then n is called a unit normal vector. Figure 4 . 1 a illustrates the normal and tangent vectors to a parametrized curve Γ . When we deal with closed curves (Figure 4 . 1 b), we will always presume that the curves are sufficiently “well-behaved” (smooth) that there is a unique inward unit normal vector and unit outward normal vectors at each point along the curve. The following Lemma formally states an observation that was made when we discussed phase portraits in Chapter 2 . Lemma 4 . 1 . 2 . Consider a system x = f ( x ) where f is continuously differentiable. Then solution trajectories in the phase portrait cannot intersect each other. Proof. Exercise. Convince yourself that if two trajectories did intersect, then this would violate the Fundamental Existence and Uniqueness Theorem 3 . 2 . 2 .
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Ordinary & Partial Differential Equations - Reynolds (2000) - Chapter 4 - Periodic, Heteroclinic

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