# 2.10 Saddles, Nodes, Foci and Centers

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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

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