Autonomous equations Autonomous systems Ordinary differential equations which do not contain the independent variable explicitly are said to be autonomous. i f i(x 1, x 2,..., x n ) for i 1,..., n As you can see there is no explicit t dependence Example 1.- Equation of motion dv 1 F (x, v) m v In this case we call and x 1 v and x 2 x. Example 2.- N-bo problem Consider N particles in a system (planetary system, for instance) d 2 x i 2 G m s (x i x s ) x i x s 5/2 s i for i 1,..., n As in Example 1 the equations are written as a first order system, i.e, introduce v i i /. Example 3.- Lorenz Attractor This is a meteorology problem studied in 1963. 3 (x y) x z + r x y dz x y z If r is large enough, behaviour is chaotic (see book of C. Sparrow). c 2009 Tecnun (University of Navarra) 1
Example 4.- The motion of the simple pendulum dv g L sin(x) v The simplest way to stu the approximate behaviour of an autonomous equation of order n when it is expressed in the form of a system of n-coupled first order equations, is illustrated in the above examples. Phase space If N 2 we can plot the solution of 1 f 1 (x 1, x 2 ) 2 f 2 (x 1, x 2 ) The solution of the system is a curve or trajectory in a 2-dimensional plane called phase plane. The trajectory is parameterized in terms of t, i.e. and. (or more generally x x(t) and y y(t)). We will assume that f 1 (x 1, x 2 ) and f 1 (x 1, x 2 ) or F (x, y) and G(x, y) The arrow indicates the increase of the Figura 1: Phase plane independent variable, t, usually called time. c 2009 Tecnun (University of Navarra) 2
Suppose that F (x, y) G(x, y) Where F and G are analytic with respect to any point (x, y ) of the (x, y) plane, i.e. F and G possess a convergent double power series. Nature of the trajectories G(x, y) F (x, y) Suppose that (x a, y a ) is such that neither F (x a, y a ) 0 nor G(x a, y a ) 0 Also assume that F (x a, y a ) 0. According to the Picard-Lindelöf theorem, the solution exists in some neighbourhood of (x a, y a ). G/F satisfies the condition of the Picard-Lindelöf theorem, i.e. locally there exists a unique solution of G(x, y) F (x, y) such that y(x a ) (x a, y a ) Similarly if F (x a, y a ) 0 then write F (x, y) G(x, y) and G(x a, y a ) 0. If F and G are not simultaneously zero, the region has plane trajectories and they do not intersect. If F (x 0, y 0 ) G(x 0, y 0 ) 0, then (x 0, y 0 ) is a critical (fixed, singular or equilibrium) point. Then F (x 0, y 0 ) 0 G(x 0, y 0 ) 0 It follows that x x 0 and y y 0 is a solution of the original equation for all t. The velocity at these points is zero so the position vector does not move. Note that while a trajectory can approach a critical point as t tends to infinity, it cannot reach such a point in a fixed time span. Suppose it were possible for a trajectory to reach a fixed point (or equilibrium) at t T, finite. Then the timereversed system of equations obtained by changing the variable t to τ and by replacing F with F and G with G, would exhibit an impossible behaviour: the position vector x(t) and y(t) would rest motionless at the critical point and then suddenly begin to move at time T, since the components of the velocity vector do not depend upon the time, they only depend on the position of the particle. c 2009 Tecnun (University of Navarra) 3
To analyse the (local) behaviour in the neighbourhood of singular points, we expand F and G in Taylor series about (x 0, y 0 ): F (x, y) a (x x 0 ) + b (y y 0 ) + a 1 (x x 0 ) 2 +... G(x, y) c (x x 0 ) + d (y y 0 ) + a 2 (x x 0 ) 2 +... There are no constant terms since F (x 0, y 0 ) G(x 0, y 0 ) 0. Hence Putting Gives Thus unless G(x, y) F (x, y) c (x x 0) + d (y y 0 ) + a 2 (x x 0 ) 2 +... a (x x 0 ) + b (y y 0 ) + a 1 (x x 0 ) 2 +... x x 0 r cos(θ) y y 0 r sin(θ) G c cos(θ) + d sin(θ) + 0(r) F a cos(θ) + b sin(θ) + 0(r) G c cos(θ) + d sin(θ) F a cos(θ) + b sin(θ) b a d c or a d b c 0 as r 0 We assume that a d b c 0. The Picard-Lindel of theorem does not apply in a neighbourhood of the critical point (x 0, y 0 ) since the function G/F is not continuous. We assume that a, b, c and d are not zero simultaneously and also assume that near (x 0, y 0 ) the solutions of the full problem are equivalent to the solutions of the linearised problem. Linearised system Defining c (x x 0) + d (y y 0 ) a (x x 0 ) + b (y y 0 ) x x x 0 gives y y y 0 c x + d y a x + b y Henceforth we shall omit the overbar, i.e. c x + d y a x + b y c 2009 Tecnun (University of Navarra) 4
The original system can be linearised by setting x a x + b y y c x + d y This linear system can be solved as was shown an earlier chapter. The characteristic polynomial of the matrix of this system is: r 2 + (a + d) r + (a d b c) 0 Denoting p a + d (trace of the matrix) and q a d b c (the determinant), we can rewrite the quadratic equation as r 2 + (p r + q 0 The eigenvalues of this system are the roots of this quadratic polynomial, i.e. r p ± p 2 4 q 2 A stu of the eigenvalues is useful to classify the fixed point. The following different cases are considered: 1. q < 0.- Since the discriminant, p 2 4 q, of the quadratic equation is always bigger than zero, we can deduce that the two roots are real numbers. As the product of them, q, is negative, they have opposite signs. The fixed point is a saddle point. 2. q > 0.- a) p 2 > 4 q.- The discriminant is positive. The eigenvalues are real having the same sign. 1) p > 0.- The eigenvalues are both positive. The critical point is an unstable node. 2) p < 0.- The eigenvalues are both negative. The critical point is a stable node b) p 2 < 4 q.- The discriminant is negative, the eigenvalues are complex conjugates. 1) p > 0.- The critical point is an unstable spiral point or spiral source. 2) p 0.- The fixed point is a centre. 3) p > 0.- The fixed point is a stable spiral point or spiral sink. There are several degenerate cases. These occur when there is a repeated eigenvalue and when an eigenvalue is zero. c) p 2 4 q c 2009 Tecnun (University of Navarra) 5
1) p > 0.- The eigenvalues are repeated and positive. The origin is an improper unstable node, if there is only one eigenvector. However, if there are two linearly independent eigenvectors, then the origin is called source point or source. 2) p < 0.- The eigenvalues are repeated and negative. The origin is an improper stable node, if there is only one eigenvector. However, if there are two linearly independent eigenvectors, then the origin is called sink point or sink. 3. q 0.- This implies that the matrix rank is one. The critical points are not isolated. We do not consider this case in our stu. Figura 2: Classification in (p,q) plane A centre may be regarded as a degenerate case. The existence of a centre depends upon the addition of two coefficients (a and d). This dependence is rather a fragile feature. Therefore, if the linear approximation to a nonlinear system predicts a centre, we cannot conclude reliably that the original system has a centre. It might be either a centre, or a stable spiral or an unstable spiral. The same applies to all degenerate (improper) cases: if they are used as linear approximation they are not reliable indicators. The following enumeration illustrates the stability or unstability of linearised autonomous systems, (the eigenvalues are denoted by r 1 and r 2 ) 1. r 1 > r 2 > 0.- Node.- Unstable. 2. r 1 < r 2 < 0.- Node.- Asymptotically stable. 3. r 1 < 0 < r 2.- Saddle point.- Unstable. 4. r 1 r 2 > 0.- Node or Improper node or spiral point.- Unstable. 5. r 1 r 2 < 0.- Node or Improper node or spiral point.- Asymptotically stable. c 2009 Tecnun (University of Navarra) 6
6. r 1 α + i β, r 2 α i β, with α > 0.- Spiral point.- Unstable. 7. r 1 α + i β, r 2 α i β, with α < 0.- Spiral point.- Asymptotically stable. 8. r 1 i β, r 2 i β.- Centre or spiral point.- Indeterminate. A precise definition of the concepts of stability and instability is now given. Let x(t) and y(t) be a given real solution of the system x F (x, y) and y G(x, y). 1. The x(t) and y(t) is Liapunov stable for t t 0 if, and only if, to each value ɛ > 0, however small, there corresponds a value δ(ɛ, t 0 ) > 0 such that [x(t 0 ) x(t 0 )] 2 + [y(t 0 ) y(t 0 )] 2 < ɛ [x(t) x(t)] 2 + [y(t) y(t)] 2 < δ for all t t, where x(t) and y(t) represent any other neighbouring solution. (ii) If the given system is autonomous, the reference to t 0 may be disregarded; the solution is either Liapunov stable, or not, for all t 0. (iii) Otherwise the solution is unstable in the sense of Liapunov. There is another definition of stability due to Poincaré. We call positive halfpath, the path in the phase plane from a particular point onwards. Let Γ be the half-path for the solution x(t) and y(t) of x F (x, y) and y G(x, y). Suppose that for every ɛ > 0 there exists such that if δ(ɛ) > 0 is the half-path starting at (x a, y a ) [x x a ] 2 + [y y a ] 2 < δ max }{{} x(t),y(t) Γ [x(t), y(t), Γ ] < ɛ Then Γ is said to be Poincaré stable. Otherwise, Γ is Poincaré unstable. If the system equations are autonomous, then Liapunov stability implies Poincaré stability. References 1. Bender, C.M. and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, McGraw-Hill International Editions, 1987. 2. Boyce, W.E. and Diprima, R.C., Elementary Differential Equations and Boundary Value Problems, 5th edition, John Wiley and Sons, 1992 3. Jordan, D.W. and Smith, P. Nonlinear Ordinary Differential Equations, Oxford University Press, 3/ed, 2004. c 2009 Tecnun (University of Navarra) 7