Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati
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....
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.
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
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.
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?
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.
We shall study the behaviour of trajectories near the equilibrium point.
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.
[ ] 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 0 1 0. [ ] 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.
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.
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.
[ ] 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.
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.
[ ] 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 1 =. 0 1 Thus B = P 1 AP = [ 0 2 2 0 ] = We consider y = By [ 0 2 2 0 ]. [ ] [ ] 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 2 1 + c 2 2, a circle.
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 2 1 + 4c 2 2, t R. The trajectories of this system lie on ellipses.
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 ***