An elementary introduction to linear dynamical system Bijan Bagchi Department of Applied Mathematics University of Calcutta E-mail : bbagchi123@rediffmail.com 18.01.2012 1 What is a dynamical system (DS)? Mathematically, a DS deals with an initial value problem of the type d−!x −! = f (t; −!x ; u) dt −! k where x denotes a vector having components (x1; x2; :::; xk) 2 R ; t is the −! −! time, f is a vector flow : f = (f1; f2; :::; fk); fug is a set of auxiliary objects −! and there exist a set of initial conditions x0 = [x1(0); x2(0); :::; xk(0)]. Thus we write 0 1 0 1 0 1 x1 f1 x1(0) B C B C B C B C B C B C B x2 C B f2 C B x2(0) C B C B C B C B C B C B C −! B : C −! B : C −! B : C x = B C, f = B C, x0 = B C. B C B C B C B : C B : C B : C B C B C B C B C B C B C B : C B : C B : C @ A @ A @ A xk fk xk(0) The case k = 1 is trivial implying a scalar equation with the solution Z t x = x0 + f(s; x; u) ds: 0 The case k = 2 has the form (with no explicit presence of t) x_i = fi(x1; x2) i = 1; 2 where we also assume fi(x1; x2) to be continuously differentiable in the neigh- borhood of [x1(0); x2(0)]. Physically, a DS addresses an evolutionary problem related to some phys- ical or chemical or biological system or lattice maps or cellular automata in which the values of variables may be specified in uniformly spaced points of time (difference equations) or with respect to their time derivatives (differ- ential equations). 2 Take the simplest canonical form given by the equation d−!x −! = −!x_ = f (−!x ) dt −! It is called autonomous when f does not depend on time directly but only −! through the state variable x . The space Rk or an appropriate subspace of dependent variables is referred to as the state space or phase space or configuration space. An autonomous system has the formal form d−!x −! = f (−!x ) dt −! −! −! where there is no dependence of t on f , f is a map Rk ! Rk and x = −! (x1; x2; :::; xk). If the map f is linear, the DS is termed a linear dynamical system. Then it has the form d−!x −! = f (−!x ) = [A]−!x ≡ A−!x dt where [A] is an k × k matrix with constant elements : Aαβ =constant, −! 1 ≤ α; β ≤ k. For k = 2, f is given by 0 1 −! −! −! a b −! f ( x ) = A x = @ A x c d Note that a second-order ODE of the general type x¨ + ax_ + bx + c = 0 can be decomposed into a set of coupled first-order ODEs : x_1 = x2 x_2 = −ax2 − bx1 − c where x1 = x and x2 =x _. Thus the k = 2 case is of much interest. 3 Definitions −! • f (t; −!x ; u) is called a vector flow or a vector field. • System of ODE is called autonomous if the vector flow does not depend on t. Otherwise the system is non-autonomous. • In the case of autonomous system and k ≤ 3, the representation of phase space is useful : dx k = 1 (x; dt ) phase plane k = 2 (x; y) phase plane k = 3 (x; y; z) phase space It is clear that if the vector flow is plotted in each point of the phase plane (k = 2), it depicts the phase portrait of the system of ODE. • x∗ is a fixed point or an equilibrium point if there is no movement in x∗ i.e. f(t; x∗; u) = 0 Example 1 Consider the damped pendulum g θ¨ + γθ_ + sinθ = 0 l With x = θ; y = θ_ it can be translated to the pair g x_ = y y_ = − sinx − γy l ) two fixed points (x = 0; y = 0) and (x = π; y = 0) and generally x = 2nπ; y = 0 and x = (2n + 1)π; y = 0. 4 Example 2 Consider a two-dimensional case in which x_1 = −x1 x_2 = −x2 The solutions are x1 = k1 exp(−t) x2 = k2 exp(−t) The fixed point is clearly the origin x1 = 0 = x2. If we eliminate t we get k1 x1 = x2 k2 ) trajectories are straight lines. What is the relevance of the fixed point with regard to the trajectories? For this we notice lim xi(t) = lim ki exp(−t) = 0 t!1 t!1 ) fixed point will be reached in the infinite future : time directions point toward the origin. Figure 1: Phase portrait 5 Dynamical configurations for k = 2 We discuss systems in two-variables of a linear space autonomous system. The general form, as noted earlier, can be written as 0 1 0 1 0 1 0 1 x_ x a11 a12 x @ A = A @ A = @ A @ A y_ y a21 a22 y where a11; a12; a21; a22 2 R. The equilibrium point for whichx _ = 0 and y_ = 0 is x = y = 0 if det(A) 6= 0. The characteristic equation is det(A − λI) = 0 I is a 2 × 2 order identity matrix 0 1 a11 − λ a12 ) det @ A = 0 a21 a22 − λ 2 ) λ − (a11 + a22)λ + (a11a22 − a12a21) = 0 ) λ2 − tr(A)λ + det(A) = 0 The eigenvalues are 1 p λ = (tr(A) ± ∆) 1;2 2 2 2 where ∆ = (tr(A)) − 4 det(A) = (a11 + a22) − 4(a11a22 − a12a21) 2 = (a11 − a22) + 4a12a21 = discriminant The different types of the dynamic behavior of this k = 2 system are dictated by the signs of the eigenvalues λ1; λ2 which in turn are controlled by the trace and determinant of A. 6 We distinguish the various features of the discriminant as follows : Figure 2: 7 Qualitative behavior of the above tendencies of λ1; λ2 Figure 3: 8 Figure 4: 9 To illustrate how the above procedure works let us consider the following example of a damped oscillator. x¨ + 2bx_ + !2x = 0 (b > 0) x_ = y y_ = −2by − !2x | {z } an equivalent coupled system Fixed point : x = 0; y = 0. 0 − λ 1 The characteristic polynomial is given by 2 −! −2b − λ Eigenvalues are determined from 0 − λ 1 p 2 2 = 0 ) λ1;2 = −b ± i ! − b 2 −! −2b − λ p 1 or from the relation λ1;2 = 2 (tr(A) ± ∆) where in this case ∆ = −4(!2 − b2) and tr(A) = −2b: We therefore summarize • an unstable node for b < −! < 0 • an unstable spiral for −! < b < 0 • a stable spiral for 0 < b < ! • a stable node for b > ! > 0 There is a change in the feature of the critical point from spiral to node at b = !. 10 General character of a linear DS : Phase plane analysis Phase plane analysis was first developed by H. Poincar´e which we will discuss in the context of studying elementary singular points. (x; y) plane ! phase plane (real domain) dx dy = P (x; y; t) = Q(x; y; t) dt dt | {z } dx generates a second order differential equation! (y = dt ) curve ! phase curve (eliminate dt) dy Q(x; y; t) = dx P (x; y; t) singular point (x0; y0) ! P (x0; y0) = 0 = Q(x0; y0) ordinary point (any) ! without the above property Interpretation : For an ordinary point there is a definite slope of the tangent to the trajectory through it. For a singular point the tangent direction is indeterminable and the trajectory degenerates into one point - the singular point itself. Cauchy's criterion : Through an ordinary point of the phase plane passes one, and only one, phase trajectory. Classical mechanics : We have typically the general solution x = x(t − t0; x0; y0) = x(t) y = y(t − t0; x0; y0) = y(t) where initial point is (x0; y0) at t = t0. Eliminate (t−t0) to get the trajectory. dx dy dy Summary : dt and dt prescribe the law of motion, dx prescribes a certain geometric curve passing through a given point. 11 Elementary singular points • Vortex point (centre) : SHM equation (attractive force) mx¨ = −kx k > 0 (stable motion) Factorized form : dx dy k = y = −( )x = −!2x dt dt m Elimination of t gives dy x = −!2 dx y x2 y2 ) + = λ (Ellipse) α2 β2 2h (α2 = β2 = 2h h integration constant) !2 x = 0 = y is the centre of ellipse and represents the singular point called the vortex point (centre). The trajectories which are closed curves can be seen to enclose the vortex point in their interior but none approaching the vortex point. 12 Figure 5: y =x _ (centre) • Saddle Point : Consider the case of repulsive force mx¨ = kx k > 0 Factorized form dx dy k = y = ( )x = !2x dt dt m Elimination of t gives dy x = !2 dx y ) y2 − !2x2 = h h integration constant (Hyperbola) h = 0 gives the asymptotes. x = 0 = y is a singular point called the saddle point or a critical point. There are two singular trajectories passing through the saddle point. Note that the motion on these two trajectories are asymptotic approaching the sad- dle point for t = +1 or t = −∞. 13 Figure 6: y =x _ (saddle point or critical point) • Focal Point : Let us focus on the damped SHM : x¨ = −!2x − 2bx_ (b2 − !2 < 0 : underdamping) It corresponds to the factorized pair dx dy = y = −2by − !2x dt dt dy (2by+!2x) implying dx = − y x = 0 = y is the singular point called the focal point. In the (x; t) variables the elementary solution is −bt x = x0e cos(¯!t + α) x0; α ! integration constants ! 0 as t ! 1 (second factor remains bounded) We are led to spirals approaching the focal point.