Department of Mathematics IIT Guwahati

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1 Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati

2 A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1, x 2,..., x n ) dx 2 dt = g 2(x 1, x 2,..., x n ) dx n = g n (x 1, x 2,..., x n ) dt Observe that the variable t DOES NOT appear explicitly on the right hand side of each differential equation in the above system. If n = 2 in the above system, then the system is called a Plane autonomous system....

3 Consider a Plane autonomous system given by: x 1(t) = g 1 (x 1, x 2 ) x 2(t) = g 2 (x 1, x 2 ) (1) A solution of the above system consists of pair of functions {x 1 (t), x 2 (t)} that satisfy (1) for all t I. The set of points { (x1 (t), x 2 (t) ) : t I} in the x 1 x 2 -plane is called a trajectory, and the x 1 x 2 -plane is referred to as the phase plane.

4 Definition: A point (a, b), where g 1 (a, b) = 0 and g 2 (a, b) = 0, is called a critical point of the system (1). If (a, b) is a critical point, then the constant functions x 1 (t) = a, x 2 (t) = b form a solution to this system. This is called an equilibrium solution. The point (a, b) also called equilibrium point. Example: Consider the system x 1(t) = x 2 (x 2 2) x 2(t) = (x 1 2)(x 2 2). The critical points are (2, 0)&(c, 2). The corresponding equilibrium solutions are x 1 (t) = 2, x 2 (t) = 0 x 1 (t) = c, x 2 (t) = 2

5 Definition: Let x 0 R n. For t 0 we define φ t (x 0 ) = x(t), where x(t) is the solution of IVP x = f(x), x(0) = x 0.

6 Definition: An equilibrium point x 0 R n of x (t) = f(x) is stable if for all ɛ > 0 there exists a δ = δ(ɛ) > 0 such that φ t ( x) x 0 < ɛ t 0 whenever x x 0 < δ. Asymptotically stable if it is stable and there exist δ > 0 such that lim φ t(x) = x 0, x such that x x 0 < δ. t If x 0 is not stable, then it is unstable. Remark: Asymptotic stability = Stability. Stability need not be Asymptotic stability always, why?

7 For the linear system, it is sufficient to study the stability of the zero solution to the homogeneous system. In the two theorems below, A denotes an n n matrix with real entries. Theorem: An equilibrium point x of the linear system x (t) = Ax(t) is stable (asymptotically stable) iff x 0 = 0 is a stable (an asymptotically stable)equilibrium point of x (t) = Ax(t). Theorem: If all eigenvalues of A have negative real parts, then the equilibrium point x 0 = 0 of x (t) = Ax(t), is asymptotically stable. However, if at least one eigenvalue of A has positive real part, then the zero equilibrium point is unstable.

8 We shall study the behaviour of trajectories near the equilibrium point.

9 Consider the linear system x (t) = Ax(t), A is a 2 2 matrix. (2) We first describe the phase portraits (set of all solution curves in the phase space R 2 ) of the linear system y (t) = By(t), P 1 AP = B (3) The phase portrait for the linear system (2) then obtained from the phase portrait for (3) under the linear transformation x = P y. Recall [ ] λ 0 if B = then the solution of IVP y 0 µ (t) = By(t) [ ] e λt 0 with y(0) = y 0 is y(t) = 0 e µt y 0.

10 [ ] λ 1 If B = then the solution of IVP y 0 λ (t) = By(t) [ ] 1 t with y(0) = y 0 is y(t) = e λt y [ ] a b If B = then the solution of IVP y b a (t) = By(t) with y(0) = y 0 is [ ] cos bt sin bt y(t) = e at y sin bt cos bt 0. We now discuss the various phase portraits that result from these solutions.

11 Case I. B = [ λ 0 0 µ ] with λ < 0 < µ. Stability of Linear Systems in R 2 The system is said to have a saddle at the origin in this case. If µ < 0 < λ, the arrows are reversed. Whenever A has two real eigenvalues of opposite sign, the phase portrait for the linear system is linearly equivalent to the phase portrait shown in Fig. 1.

12 Case II. B = Stability of Linear Systems in R 2 [ λ 0 0 µ ] with λ µ < 0 or B = [ λ 1 0 λ ] with λ < 0. In each of these cases, the origin is refereed to as a stable node. It is called a proper node in the first case with λ = µ and an improper node in the other two cases. If λ µ > 0 or if λ > 0 in Case II, the arrows are reversed and the origin is referred to as an unstable node. The stability of the node is determined by the sign of the eigenvalues: stable if λ µ < 0 and unstable if λ µ > 0.

13 [ ] a b Case III. B = with a < 0. b a Stability of Linear Systems in R 2 The origin is refereed to as a stable focus in these cases. If a > 0, the trajectories spiral away from the origin with increasing t. The origin is called an unstable focus. Whenever A has a pair of complex conjugate eigenvalues with nonzero real part, a ± ib, with a < 0, the phase portraits for the system (3) is linearly equivalent to one of the phase portraits shown above.

14 Case IV. B = Stability of Linear Systems in R 2 [ 0 b b 0 ]. The system (3) is said to have a center at the origin in this case. Whenever A has a pair of purely imaginary complex conjugate eigenvalues, ±ib, the phase portrait of the linear system (2) is linearly equivalent to one of the phase portraits shown above. Note that trajectories or solution curves lie on circles x(t) = constant. In general, the trajectories of the system (2) will lie on ellipse.

15 [ ] 0 4 Example: Consider the system x (t) = Ax(t), A =. 1 0 Note that A has eigenvalues λ = ±2i. The matrix P and P 1 are [ ] [ ] 2 0 1/2 0 P = and P =. 0 1 Thus B = P 1 AP = [ ] = We consider y = By [ ]. [ ] [ ] cos(2t) sin(2t) y(t) = e Bt c1 c = sin(2t) cos(2t) c 2 y 1 (t) = c 1 cos(2t) c 2 sin(2t) y 2 (t) = c 1 sin(2t) + c 2 cos(2t). y 2 1(t) + y 2 2(t) = c c 2 2, a circle.

16 The solution x(t)is then given by x = P y, x 1 (t) = c 1 cos(2t) 2c 2 sin(2t) x 2 (t) = 1 2 c 1 sin(2t) + c 2 cos(2t). Observe that x 2 1(t) + 4x 2 2(t) = c c 2 2, t R. The trajectories of this system lie on ellipses.

17 Real eigenvalues Saddle point λµ < 0 Unstable Node λµ > 0 stable if λ < 0, µ < 0 Unstable if λ > 0, µ > 0 Complex eigenvalues Focus Re λ 0 Stable if Reλ < 0 Unstable if Reλ > 0 Center Reλ = 0 Stable Note: We consider the case A is invertible, λ 0, µ 0. *** End ***