Types of critical points Def. (a, b) is a critical point of the autonomous system Math 216 Differential Equations Kenneth Harris kaharri@umich.edu Department of Mathematics University of Michigan November 7, 28 x = F(x, y), y = G(x, y) when F(a, b) = and G(a, b) =. Classification. There are two dimensions to classifying critical points Are trajectories drawn to or repulsed from a critical point? How do the trajectories approach the critical point? There are several types of critical points. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable) 3 Spiral (stable or unstable) 4 Center (always stable) 5 Saddle point (always unstable) Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 1 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 3 / 1 Stability Def. A critical point (a, b) is stable provided all points sufficiently close to (a, b) remain close to it. It is asymptotically stable if all points are drawn to it. Example of a stable equilibrium Example. (, ) is a sink (so, stable). There are four kinds of stability. A center is stable, but not asymptotically stable. A sink is asymptotically stable. A source is unstable and all trajectories recede from the critical point. A saddlepoint is unstable, although some trajectories are drawn to the critical point and other trajectories recede. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 4 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 5 / 1
Example of an unstable equilibrium Example of a stable equilibrium Example. (, ) is a source (so unstable). Example. (, ) is an center (so stable). Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 6 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 7 / 1 Example: Saddle point Example. (, ) is an saddle point (so unstable). How do trajectories approach or recede? Nodes and spiral points Definition. A critical point (a, b) is a node if Every trajectory approaches (a, b) as t or every trajectory recedes from (a, b) as t, and Each trajectory approaches (or recedes) from (a, b) in a fixed direction. (That is, every trajectory is tangent to a line through (a, b).) Three types of approach to critical points. A critical point is a proper node if trajectories approach or recede in all directions. A critical point is an improper node if all trajectories approach or recede in just two directions. A critical point is a spiral point if trajectories spiral around the critical point as they approach or recede. A spiral point cannot be a node!! Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 8 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 9 / 1
Example: A proper node Example. (, ) is a proper node which is stable. Example: A proper node Example. (, ) is a improper node which is stable. (Trajectories approach in only two directions.) Eigenvector solutions in red. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 1 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 11 / 1 Example of an unstable spiral point Types of critical points Example. (, ) is an unstable spiral point. Summary. There are several types of critical points based on stability and how solutions approach. 1 Proper node (stable or unstable) 2 Improper node (stable or unstable) 3 Spiral (stable or unstable) 4 Center (always stable, but not asymptotically stable) 5 Saddle point (always unstable) Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 12 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 13 / 1
Critical points of linear systems Linear systems. We will study the properties at critical points of 2 2 linear systems [ ] [ ] [ ] x a b x y = c d y where ad bc (so, the matrix is invertible). Critical point. There is only one critical point at (, ). Let λ 1 and λ 2 be eigenvalues. Five cases. 1 λ 1 and λ 2 are distinct with the same sign. 2 λ 1 and λ 2 are distinct with the different signs. 3 λ 1 = λ 2. 4 λ 1 and λ 2 are complex conjugates with nonzero real parts. 5 λ 1 and λ 2 are pure imaginary. Real eigenvalues Real eigenvalues. When λ 1 and λ 2 are real and distinct, solutions take the form x(t) = c 1 v 1 e λ 1t + c 2 v 2 e λ 2t where v 1 and v 2 are the associated eigenvectors. c 1 = : the solution takes the form x(t) = c 2 v 2 e λ 2t x(t) travels along the eigenvector v 2. c 2 = : the solution takes the form x(t) = c 1 v 1 e λ 1t x(t) travels along the eigenvector v 1. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 15 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 16 / 1 Case 1. Distinct, real, same sign Case 1. When < λ 1 < λ 2 : solutions have the form Case 1. Distinct, real, same sign < λ 1 < λ 2 : [ ] 1 2 x(t) = c 1 v 1 e λ 1t + c 2 v 2 ( e λ 1 t ) k where k = λ 2 λ 1 > 1 Stability. x(t) as t. The origin is a source (so stable). Type. Trajectories are drawn to v 2. That is, trajectories not starting on v 1 approach v 2 at t. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 17 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 18 / 1
Case 1. Distinct, real, same sign Case 1. When λ 1 < λ 2 < : solutions have the form Case 1. Distinct, real, same sign λ 1 < λ 2 < : [ ] 1 2 x(t) = c 1 v 1 ( e λ 2 t ) k + c2 v 2 e λ 2t where k = λ 2 λ 1 > 1 Stability. x(t) as t. The origin is a sink (so stable). Type. Trajectories are drawn to v 2. That is, trajectories not starting on v 1 approach v 2 at t. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 19 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 2 / 1 Case 2. Distinct, real, opposite signs Case 2. When λ 1 < < λ 2 : solutions have the form Case 2. Distinct, real, opposite signs λ 1 < < λ 2 : [ ] 1 2 x(t) = c 1 v 1 e λ 1t + c 2 v 2 e λ 2t where v 1 and v 2 are the associated eigenvectors. Stability. The origin is a saddlepoint. x(t) at t when c 2. x(t) at t when c 2 =. Type. Eigenvectors are drawn to v 2. That is, trajectories not starting on v 1 approach v 2 at t. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 21 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 22 / 1
Case 3. Multiplicity two Case 3. λ 1 = λ = λ 2. Two eigenvectors. The solution has the form Case 3. Multiplicity two Two eigenvectors. [ ] 1 1 x(t) = ( c 1 v 1 + c 2 v 2 ) e λt Stability. λ > : x(t) as t. The origin is a source. λ < : x(t) as t. The origin is a sink. Type. The trajectory is along the line c 1 v 1 + c 2 v 2. The origin is a proper node. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 23 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 24 / 1 Case 3. Multiplicity two Case 3. λ 1 = λ = λ 2. One eigenvector. The solution has the form x(t) = c 1 v 1 e λt + c 2 ( v1 t + v 2 ) e λt = ( (c 2 t + c 1 )v 1 + c 2 v 2 ) e λt where v 1 is the eigenvector associated with λ, and v 2 is a linearly independent vector satisfying (A λi)v 2 = v 1. (See Section 5.4). Case 3. Multiplicity two One eigenvector. [ ] 1 1 1 Stability. λ > : x(t) as t. The origin is a source. λ < x(t) as t. The origin is a sink. Type. The trajectory is along (c 2 t + c 1 )v 1 + c 2 v 2, so is drawn to v 1. The origin is an improper node. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 25 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 26 / 1
Case 4. Complex Case 4. λ 1 and λ 2 are complex conjugates with nonzero real parts. Let the eigenvalues be p ± qi and eigenvectors a ± ib. A general solution: x(t) = e pt( c 1 x 1 (t) + c 2 x 2 (t) ) where x 1 (t) = a cos qt + b sin qt and x 2 (t) = b cos qt a sin qt. So, x 1 and x 2 are periodic: x 1 (t) = x 1 (t + 2π q ) and x 2(t) = x 2 (t + 2π q ) Case 4. Complex λ 1, λ 2 = p ± qi, p >. [ 1 ] 2 2 1 λ = 1 ± i2 Stability and Type. p > : x(t) as t. Origin is a spiral source, p < : x(t) as t. Origin is a spiral sink. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 27 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 28 / 1 Case 4. Complex λ 1, λ 2 = p ± qi, p <. [ 1 ] 2 2 1 λ = 1 ± 2i Case 5. Pure Imaginary Case 5. λ 1 and λ 2 are pure imaginary. Let the eigenvalues be ±qi and eigenvectors a ± ib. A general solution: x(t) = c 1 x 1 (t) + c 2 x 2 (t) where x 1 (t) = a cos qt + b sin qt and x 2 (t) = b cos qt a sin qt. So, x 1 and x 2 are periodic: x 1 (t) = x 1 (t + 2π q ) and x 2(t) = x 2 (t + 2π q ) Stability and Type. The origin is a center. Trajectories are ellipses. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 29 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 3 / 1
Case 5. Imaginary λ 1, λ 2 = p ± qi, p >. [ ] 1 1 λ = ±i Summary λ 1, λ 2 are eigenvalues of the linear system. Type and stability of critical point. λ 1 < λ 2 < : Stable improper node. λ 1 = λ 2 < : Stable node λ 1 < < λ 2 : Unstable saddle point λ 1 = λ 2 > : Unstable node λ 1 > λ 2 > : Unstable improper node λ 1, λ 2 = p ± iq, p < : Spiral sink λ 1, λ 2 = p ± iq, p > : Spiral source λ 1, λ 2 = ±iq: center Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 31 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 32 / 1 Computing eigenvalues of 2 2 matrices The characteristic polynomial for the real-valued matrix [ ] a b A = c d is a λ c b d λ = (a λ)(d λ) bc = λ 2 (a + d)λ + (ad bc) = λ 2 tr(a) + det(a) where tr(a) = a + d is the sum of the diagonal elements of A. The eigenvalues are tr(a) 2 ± 1 tr(a) 2 2 4 det(a). Perturbing a matrix Perturbations. Consider a small perturbation of the elements of A: [ A a b = ] c where a, b, c, d are close to a, b, c, d. Eigenvalues are where tr(a ) 2 ± 1 2 d tr(a ) 2 4 det(a ). tr(a ) = a + d is close to tr(a) = a + d det(a ) = a d b c is close to det(a) = ad bc Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 34 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 35 / 1
Eigenvalues under perturbation Eigenvalues under perturbation Eigenvalues of A: tr(a) 2 ± 1 2 tr(a)2 4 det(a). Assumption 1. Suppose tr(a) 2 4 det(a). Then, tr(a ) 2 4 det(a ), with the same sign. Assumption 2. Suppose tr(a). Then, tr(a ), with the same sign. Stability and Type. Two cases λ = p ± iq is a complex eigenvalue of A, then λ = p ± iq is an eigenvalue A where p, p have the same sign. A and A have the same qualitative properties at (, ). λ 1, λ 2 are distinct real eigenvalues for A, then the corresponding eigenvalues λ 1, λ 2 for A are both real and have the same signs. A and A have the same qualitative properties at (, ). Eigenvalues of A: tr(a) 2 ± 1 2 tr(a)2 4 det(a). Assumption. Suppose A has one eigenvalue. Then, tr(a), so tr(a ), with the same sign. However, tr(a) 2 4 det(a) =. Stability and Type. Three possibilities 1 tr(a ) 2 4 det(a ) =. Same qualitative properties at (, ). 2 tr(a ) 2 4 det(a ) >. The eigenvalues of A are distinct reals with the same sign. 3 tr(a ) 2 4 det(a ) <. Then eigenvalues of A are complex and (, ) is a spiral point with the same stability as that of A. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 36 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 37 / 1 Eigenvalues under perturbation Summary Eigenvalues of A: tr(a) 2 ± 1 2 tr(a)2 4 det(a). Assumption. Suppose A has imaginary eigenvalues. Then, tr(a) 2 4 det(a) <, and this remains true for A. However, tr(a) =. Stability and Type. Three possibilities 1 tr(a) =. No change in the qualitative properties at (, ). 2 tr(a) >. Then (, ) is now a spiral source. 3 tr(a) <. Then (, ) is now a spiral sink. Theorem Let λ 1 and λ 2 be the eigenvalues of a matrix A, and A a sufficiently small perturbation of A. Then, the qualitative properties of the critical point (, ) for A satisfies 1 If λ 1 = λ 2, then (, ) of A is either a node, or a spiral point. It is asymptotically stable if λ 1 < and unstable if λ 1 >. 2 If λ 1 and λ 2 are pure imaginary, then (, ) is either a center or a spiral. It could be any of asymptotically stable, stable, or unstable. 3 Otherwise, (, ) has the same type and stability at A as at A. Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 38 / 1 Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 39 / 1
Summary Distribution of critical points in the Trace-Determinant plane. [ ] a b A =, T = tr(a) = a + d, D = det(a) = ad bc. c d Sensitive areas: Places where type of critical point sensitive to perturbations. D Spiral Sink Spiral Source center stable Nodal Sink T^2 4D Nodal Source T Kenneth Harris (Math 216) Math 216 Differential Equations November 7, 28 4 / 1