2.10 Saddles, Nodes, Foci and Centers

Transcription

1 2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one of the phase portraits in Figures 1-4 in Section 1.5; i.e., if there exists a nonsingular linear transformation which reduces the matrix A to one of the canonical matrices B in Cases I-IV of Section 1.5 respectively. In Section 2.6, a nonlinear system is ẋ = f(x (2 was said to have a saddle, a sink or a source at a hyperbolic equilibrium point x 0 if the linear part of f at x 0 had eigenvalues with both positive and negative real parts, only had eigenvalues with negative real parts, or only had eigenvalues with positive real parts, respectively. In this section, we define the concept of a topological saddle for the nonlinear system (2 with x R 2 and show that if x 0 is a hyperbolic equilibrium point of (2, then it is a topological saddle if and only if it is a saddle of (2; i.e., a hyperbolic equilibrium point x 0 is a topological saddle for (2 if and only if the origin is a saddle for (1 with A = Df(x 0. We discuss topological saddles for nonhyperbolic equilibrium points of (2 with x R 2 in the next section. We also refine the classification of sinks of the nonlinear system (2 into stable nodes and foci and show that, under slightly stronger hypotheses on the function f, a hyperbolic critical point x 0 is a stable node or focus for the nonlinear system (2 if and only if it is respectively a stable node or focus for the linear system (1 with A = Df(x 0. Similarly, a source of (2 is either an unstable node or focus of (2 as defined below. Finally, we define centers and center-foci for the nonlinear system (2 and show that, under the addition of nonlinear terms, a center of the linear system (1 may become either a center, a center-focus, or a stable or unstable focus of (2. Before defining these various types of equilibrium points for planar systems (2, it is convenient to introduce polar coordinates (r, θ and to rewrite the system (2 in polar coordinates. In this section we let x = (x, y T, f 1 (x = P (x, y and f 2 (x = Q(x, y. The nonlinear system (2 can then be written as ẋ = P (x, y, ẏ = Q(x, y. If we let r 2 = x 2 + y 2 and θ = tan 1 (y/x, then we have and rṙ = xẋ + yẏ r 2 θ = xẏ yẋ. It follows that for r > 0, the nonlinear system (3 can be written in terms of polar coordinates as or dr dθ ṙ = P (r cos θ, r sin θ cos θ + Q(r cos θ, r sin θ sin θ r θ = Q(r cos θ, r sin θ cos θ P (r cos θ, r sin θ sin θ = r[p (r cos θ, r sin θ cos θ + Q(r cos θ, r sin θ sin θ] Q(r cos θ, r sin θ cos θ P (r cos θ, r sin θ sin θ. (5 Writing the system of differential equations ( 3 in polar coordinates will often reveal the nature of the equilibrium point or critical point at the origin. We assume that x 0 R 2 is an isolated equilibrium point of the nonlinear system (3 which has been translated to the origin; r(t, r 0, θ 0 and θ(t, r 0, θ 0 will denote the solution of the nonlinear system (4 with r(0 = r 0 and θ(0 = θ 0. (3 (4

2 2 Definition. The origin is called a center for the nonlinear system (2 if there exists a δ > 0 such that every solution curve of (2 in the deleted neighborhood N δ (0 {0} is a closed curve with 0 in its interior. Consider ẋ = y xy, ẏ = x + x 2. Let us write the system in polar coordinates. For r > 0, we have ṙ = 0, θ = 1 + x. Thus θ > 0 for x > 1. So, along any trajectory of this system in the half plane x > 1, r(t is constant and θ(t increases without bound as t. Thus the origin is a center for this nonlinear system. Definition. The origin is called a center-focus for (2 if there exists a sequence of closed solution curves Γ n, with Γ n+1 in the interior of Γ n such that Γ n 0 as n and such that every trajectory between Γ n and Γ n+1 spirals toward Γ n or Γ n+1 as t ±. ẋ = y + x x 2 + y 2 sin(1/ x 2 + y 2, ẏ = x + y x 2 + y 2 sin(1/ x 2 + y 2, for x 2 + y 2 0 and define f(0 = 0. In polar coordinates we have ṙ = r 2 sin(1/r, θ = 1 for r > 0 and ṙ = 0 for r = 0. Clearly, ṙ = 0 for r = 1/(nπ; i.e., each of the circles r = 1/(nπ is a trajectory of this system. Furthermore, for nπ < 1/r < (n + 1π, ṙ < 0 if n is odd and ṙ > 0 if n is even; i.e., the trajectories between the circles r = 1/(nπ spiral inward or outward to one of these circles. Thus, we see that the origin is a center-focus for this nonlinear system. Definition. The origin is called a stable focus for (2 if there exists a δ > 0 such that for 0 < r 0 < δ and θ R, r(t, r 0, θ 0 0 and θ(t, r 0, θ 0 as t. It is called an unstable focus if r(t, r 0, θ 0 0 and θ(t, r 0, θ 0 as t. Any trajectory of (2 which satisfies r(t 0 and θ(t as t ± is said to spiral toward the origin as t ±. Consider In polar coordinates, for r > 0, we have ẋ = y x 3 xy 2, ẏ = x y 3 x 2 y. ṙ = r 3, θ = 1. Thus for t > 1/(2r 0 2 and We see that r(t = r 0 (1 + 2r 2 0t 1/2 θ(t = θ 0 + t. r(t 0

3 3 and θ(t as t. Thus the origin is a stable focus for this nonlinear system. Example 2. Consider In polar coordinates, for r > 0, we have Thus for t < 1/(2r 0 2 and We see that and ẋ = y + x 3 + xy 2, ẏ = x + y 3 + x 2 y. ṙ = r 3, θ = 1. r(t = r 0 (1 2r 2 0t 1/2 θ(t = θ 0 + t. r(t 0 θ(t as t. Thus the origin is an unstable focus for this nonlinear system. Definition. The origin is called a stable node for (2 if there exists a a δ > 0 such that for 0 < r 0 < δ and θ R, r(t, r 0, θ 0 0 as t and lim θ(t, r 0, θ 0 exists; i.e., each trajectory in a deleted t neighborhood of the origin approaches the origin along a well-defined tangent line as t. The origin is called an unstable node if r(t, r 0, θ 0 0 and lim θ(t, r 0, θ 0 exists for all r 0 (0, δ t and θ R. The origin is called a proper node for (2 if it is a node and if every ray through the origin is tangent to some trajectory of (2. Example 3. Consider ( ẋ ẏ = ( ( x y The only critical point is (0, 0. The eigenvalues are α 1 = 4 and α 2 = 3 with corresponding eigenvectors ( ( 1 2,. 1 1 As a result the general solution is. x(t = c 1 e 4t + 2c 2 e 3t, y(t = c 1 e 4t + c 2 e 3t. r(t = ( y(t x 2 (t + y 2 (t 0 as t and lim θ(t, r 0, θ 0 = lim tan 1 t t x(t The critical point (0, 0 is a stable node. = tan 1 ( 1 2 exists. Definition. The origin is a (topological saddle for (2 if there exist two trajectories Γ 1 and Γ 2 which approach 0 as t and two trajectories Γ 3 and Γ 4 which approach 0 as t and if there exists a δ > 0 such that all other trajectories which start in the deleted neighborhood of the origin N δ (0 {0} leave N δ (0 as t ±. The special trajectories Γ 1,, Γ 4 are called separatrices.

4 4 For a (topological saddle, the stable manifold at the origin S = Γ 1 Γ 2 {0} and the unstable manifold at the origin U = Γ 3 Γ 4 {0}. If the trajectory Γ i approaches the origin along a ray making an angle θ i with the x axis where θ i ( π, π] for i = 1,..., 4, then θ 2 = θ 1 ± π and θ 4 = θ 3 ± π. Example 4. Consider ( ẋ ẏ = ( ( x y In this case the eigenvalues are λ = ±1. It has the general solution The trajectories are Let us show that the origin is a saddle point. Let. x(t = c 1 e t + c 2 e t, y(t = c 1 e t c 2 e t. x 2 y 2 = 4c 1 c 2. Γ 1 : x 1 = e t, y 1 = e t Γ 2 : x 2 = e t, y 2 = e t Γ 3 : x 3 = e t, y 3 = e t If c 1 c 2 0, then the trajectories are hyperbolas. Γ 4 : x 4 = e t, y 4 = e t Theorem 1. Let E be an open subset of R 2 containing the origin and let f C 1 (E. If the origin is a hyperbolic equilibrium point of the nonlinear system ẋ = f(x, then the origin is a (topological saddle for this nonlinear system if and only if the origin is a saddle for the linear system with A = Df(0. ẋ = x + 2y + x 2 y 2, ẏ = 3x + 4y 2xy. [ ] 1 2 Then A = Df(0 = has eigenvalues λ 3 4 1,2 = 5 ± 33. Thus (0, 0 is a saddle point for the 2 linear system ẋ = Ax and hence (0, 0 is a (topological saddle for this nonlinear system. The next example, due to Perron, shows that a node for a linear system may change to a focus with the addition of nonlinear terms.

5 5 Example 2. ẋ = x ẏ = y + y ln x, 2 +y 2 x ln x, 2 +y 2 for x 2 + y 2 0 and define f(0 = 0. In polar coordinates we have ( Thus, r(t = r 0 e t and θ(t = θ 0 ln 1 t ṙ = r, θ = 1 ln r. ln r 0. We see that for r 0 < 1, r(t 0 and θ(t as t and therefore the origin is a stable focus for this nonlinear system; however, it is a stable proper node for the linearized system. This example shows that the hypothesis f C 1 (E is not strong enough to imply that the phase portrait of a nonlinear system ẋ = f(x is diffeomorphic to the phase portrait of its linearization. Theorem 2. Let E be an open subset of R 2 containing the origin and let f C 2 (E. Suppose that the origin is a hyperbolic critical point of ẋ = f(x Then the origin is a stable (or unstable node for this nonlinear system if and only if it is a stable (or unstable node for the linear system with A = Df(0. And the origin is a stable (or unstable focus for the nonlinear system ẋ = f(x if and only if it is a stable (or unstable focus for the linear system with A = Df(0. ẋ = y + x x 2 + y 2 sin(1/ x 2 + y 2, ẏ = x + y x 2 + y 2 sin(1/ x 2 + y 2, for x 2 + y 2 0 and define f(0 = 0. In polar coordinates we have ṙ = r 2 sin(1/r, θ = 1 for r > 0 and ṙ = 0 for r = 0. Clearly, ṙ = 0 for r = 1/(nπ; i.e., each of the circles r = 1/(nπ is a trajectory of this system. Furthermore, for nπ < 1/r < (n + 1π, ṙ < 0 if n is odd and ṙ > 0 if n is even; i.e., the trajectories between the circles r = 1/(nπ spiral inward or outward to one of these circles. Thus, we see that the origin is a center-focus for this nonlinear system. Theorem 3. Let E be an open subset of R 2 containing the origin and let f C 1 (E with f(0 = 0. Suppose that the origin is a center for the linear system with A = Df(0. Then the origin is either a center, a center-focus or a focus for the nonlinear system ẋ = f(x.

6 6 ẋ = y x x 2 + y 2, ẏ = x y x 2 + y 2. The origin is a center for the linear system ẋ = Df(0x. In polar coordinates we have ṙ = r 2, θ = 1. Thus the solution is given by r(t = r r 0 t, θ(t = θ 0 + t. We see that for r 0 > 0, r(t 0 and θ(t as t and therefore the origin is a stable focus for this nonlinear system. Corollary 1. Let E be an open subset of R 2 containing the origin and let f be analytic in E with f(0 = 0. Suppose that the origin is a center for the linear system with A = Df(0. Then the origin is either a center or a focus for the nonlinear system Definition. ( Symmetries The system ẋ = f(x. ẋ = P (x, y, ẏ = Q(x, y, is said to be symmetric with respect to thex axis if it is invariant under the transformation (t, y ( t, y; it is said to be symmetric with respect to the y axis if it is invariant under the transformation (t, x ( t, x. Consider ẋ = y xy, ẏ = x + x 2. It is symmetric with respect to the x-axis, but not with respect to the y-axis. Theorem 4. Let E be an open subset of R 2 containing the origin and let f C 1 (E with f(0 = 0. If the nonlinear system ẋ = f(x is symmetric with respect to the x axis or the y axis, and if the origin is a center for the linear system with A = Df(0, then the origin is a center for the nonlinear system ẋ = f(x. Problems: 1,4