MTH5123 Differential Equations Lecture Notes Week 11 PDF

MTH5123 Differential Equations Lecture Notes Week 11 School of Mathematical Sciences Queen Mary University of London 5 Stability of Solutions of ODEs The subject of stability studies is to understand how a change of initial conditions or a change i...

MTH5123 Differential Equations Lecture Notes Week 11 School of Mathematical Sciences Queen Mary University of London 5 Stability of Solutions of ODEs The subject of stability studies is to understand how a change of initial conditions or a change in parameter values of equations defining a dynamical system (e.g., coefficients in front of derivatives) affects the behaviour of the solutions, especially when the indepen- dent variable (usually interpreted as time t) tends to infinity, t → ∞. The main goal is to establish criteria ensuring that the solution will change only slightly if a small (in an appropriate sense) change in the initial conditions or parameters is implemented. This type of question is of great importance for practical applications, as parameters of differential equations which govern real-life processes as, e.g., the functioning of mechanical aggregates or electronic devices, are known only approximately due to unpredictable changes in tem- perature, humidity or other properties of the environment. We will only discuss the stability of solutions of systems of two coupled first-order ODEs, but the basic ideas can be extended to any number of equations. As usual we will use vector notation by writing down a system of two general non-autonomous ODEs in normal form as     y1 f1 (t, y1 , y2 ) ẏ = f (t, y) , y = ,f=. (5.1) y2 f2 (t, y1 , y2 ) Stability theory is mainly based on the following two definitions: Definition: Lyapunov stability   a1 A solution y∗ (t) of (5.1) corresponding to the initial condition y∗ (0) = a1 = is called b1 Lyapunov stable (or simply stable) if for any (arbitrarily   small)  > 0 we can find a δ > 0 a2 such that if another initial condition y(0) = a2 = is chosen inside a circle of radius b2 δ around the initial point y∗ (0) then for any time t > 0 the solution y(t) corresponding to the initial condition y(0) 1. exists, and 2. will stay inside a ”tube” of radius  around the solution y∗ (t), see Fig. (5.1). In mathematical shortcut notation this definition reads ∀ > 0 ∃δ > 0 s.t. ∀t > 0 |y(0) − y∗ (0)| < δ ⇒ |y(t) − y∗ (t)| < . Definition: asymptotic stability The solution y∗ (t) of (5.1) corresponding to the initial condition y∗ (0) = a is called asymp- totically stable if it is 1. Lyapunov stable, and  3 Figure 5.1: A sketch of the stability tube according to the definition of Lyapunov stability. y 2. there exists a δ > 0 such that the condition |y(0)−y∗ (0)| < δ implies |y(t)−y∗ (t)| → 0 for t → ∞. y Note: 1. Conditions 1. and 2. are independent, i.e., neither does 1. imply 2., nor does 2. imply 1. For the first direction see the counterexample below; the second direction is less obvious, but there are also counterexamples. 2. Making in (5.1) the change of variables z(t) ≡ y(t)−y∗ (t) one can show that investigating the stability of any solution y∗ (t) of the system (5.1) can always be reduced to investigating the stability of the zero solution z(t) = 0 of the transformed system ż = f (t, z); see our previous discussion at the beginning of Section 4.1.1. Reformulating the definition of (Lyapunov) stability in terms of z(t) we arrive at the fol- lowing expression: Stability of the zero solution z = 0, i.e., of the fixed point at z = 0, means that for any  > 0 we can find a δ > 0 such that ∀t > 0 |z(0)| < δ implies |z(t)| < . The definition of asymptotic stability can be reformulated accordingly. From now on we will concentrate on stability of the zero solution only. Example:     y1 0 Is the zero solution = of the system y˙1 = −4y2 , y˙2 = y1 (Lyapunov) stable? y2 0 Is it asymptotically stable? Solution:   0 −4 The system can be written as ẏ = Ay with A =. The eigenvalues are purely 1 0   2 imaginary, λ1 = 2i, λ2 = −2i, and the associated eigenvectors are u1,2 =. According ∓i to (4.19) the general solution of this system is given by     2it 2 −2it 2 y(t) = c1 e + c2 e. −i i 4 5 Stability of Solutions of ODEs Determining the two constants c1 , c2 by imposing the initial conditions y1 (0) = a, y2 (0) = b and expressing the solution in terms of real functions yields a y1 (t) = a cos (2t) − 2b sin (2t) , y2 (t) = sin (2t) + b cos (2t). 2 As we have seen before, trajectories of this type are ellipses, y12 + 4y22 = a2 + 4b2. But this implies: Given any  > 0 let us choose δ = /2. Then by choosing the initial conditions (a, b) to be inside a circle of radius δ, that is, a2 + b2 < δ 2 = 2 2 p /4, we find that y (t) = y12 + y22 < y12 + 4y22 = a2 + 4b2 < 4(a2 + b2 ) < 2 , hence |y| = y12 + y22 <  for any time t. We have thus shown that the zero solution y1 = y2 = 0 is Lyapunov stable. However, it is not asymptotically stable, as each solution rotates around its ellipse without approaching the origin for t → ∞, see Fig. 5. Figure 5.2: Sketch of the stability of the zero solution for elliptic trajectories providing an example that Lyapunov stability does not imply asymptotic stability. 5.1 Stability criteria for systems of two first-order linear ODEs with constant coefficents Our goal is to formulate the stability conditions for the fixed point at y1 = y2 = 0 of any system of the form     y˙1 y1 =A , (5.2) y˙2 y2 where the matrix A is time independent. We will furthermore assume that A is characterized by distinct eigenvalues λ1 6= λ2. One can then prove the following statement: Theorem: Define s ≡ max {Reλ1 , Reλ2 }. Then the zero solution y = 0 of (5.2) is 1. unstable for s > 0, 2. stable for s = 0, and 3. asymptotically stable for s < 0. 5.2 Lyapunov function method for investigating stability 5 Instead of providing a proof we just outline the basic idea of it: If s > 0 then est → ∞ for t → +∞. Hence at least one of the factors |eλ1 t |, |eλ2 t | (or both) grows without bound in time, and the modulus of the solution must increase as well implying instability. Similarly, if s < 0 then est → 0 for t → +∞ implying that both |eλ1 t |, |eλ2 t | → 0 as t → ∞. This means that the modulus of the solution must vanish asymptotically for t → ∞ so that system is asymptotically stable. Finally, for s = 0 the eigenvalues are purely imaginary and complex conjugate. We know that in this case the trajectories are ellipses that neither approach zero nor go to infinity. Instead, they remain at a finite distance from the origin, hence this case is stable but not asymptotically stable. For a proof these ideas need to be formalized in terms of equations by starting from the general solution (4.19) for the general initial value problem y1 (0) = a, y2 (0) = b. Note: Although we considered only the case of distinct eigenvalues λ1 6= λ2 the theorem can be generalized to λ1 = λ2 = λ showing that also in this case the zero solution is unstable for λ > 0, stable for λ = 0 and asymptotically stable for λ < 0. 5.2 Lyapunov function method for investigating stability Consider again the general system of two first order ODEs written in normal form (5.1). Suppose that y(t) is a solution of (5.1). Then for any continuously differentiable function V (y) defined on the same domain as y(t) one can define its values at any moment of time t on the solution y(t) as v(t) ≡ V (y1 (t), y2 (t)). We will need the expression for the time derivative v̇ = dv dt of such a function, which by using the chain rule of differentiation can be obtained to ∂V ∂V v̇ = y˙1 + y˙2. (5.3) ∂y1 ∂y2 Using (5.1) in the form of y˙1 = f1 (t, y1 , y2 ), y˙2 = f2 (t, y1 , y2 ) we obtain ∂V ∂V v̇ = f1 (t, y1 , y2 ) + f2 (t, y1 , y2 ) ≡ Df (V ) , (5.4) ∂y1 ∂y2 where we introduced the notation Df (V ). In Calculus II you have learned that this equation defines the directional derivative of V along f. Within our specific context, the above expression is called the orbital derivative. Note that Df (V ) is determined for any value of y solely by the functional form of V (y) and the form of the right-hand side of the system (5.1) without the need to know the explicit solution of the latter system. Df (V ) can be used to formulate the following statement, which we give without proof: Theorem: Lyapunov Stability Theorem Let y(t) = 0 be a solution of (5.1) and assume that inside the circle 0 < |y| < R there exists a continuously differentiable function V (y) satisfying 1. V (y = 0) = 0 2. V (y 6= 0) > 0 3. The derivative of V along f is non-positive, Df (V ) ≤ 0 for (y1 , y2 ) 6= (0, 0). 6 5 Stability of Solutions of ODEs Then the zero solution y(t) = 0 is stable. The function V (y) featuring in this theorem is called the Lyapunov function of the system (5.1). While such a function can be found for certain classes of differential equations, see the following example, unfortunately there does not exist a systematic way of how to construct it for a given system of differential equations. Note: If the third condition is replaced by Df (V ) < 0 being strictly negative one can prove that the zero solution y(t) = 0 is asymptotically stable, which is called the Lyapunov Asymp- totic Stability Theorem. Example: Verify that the function V (y1 , y2 ) = y12 + y22 is a valid Lyapunov function for the system y˙1 = −y2 − y13 , y˙2 = y1 − y23. Is the zero solution asymptotically stable? Solution: V (y1 , y2 ) satisfies the first and the second condition in the Lyapunov Stability Theorem. For any (y1 , y2 ) 6= (0, 0) we have ∂V ∂V y˙2 = 2y1 −y2 − y13 + 2y2 (y1 − y23 ) = −2(y14 + y24 ) < 0
Df (V ) = y˙1 + ∂x ∂y so that the zero solution is not only stable but even asymptotically stable. Theorem: Consider a continuously differentiable function V (y) satisfying 1. V (y = 0) = 0 2. V (y 6= 0) > 0 If the system (5.1) is autonomous and can be written as ∂V ẏ1 = − (5.5) ∂y1 ∂V ẏ2 = − (5.6) ∂y2 then the dynamical system is called a gradient flow, y = 0 is an equilibrium solution of the dynamical system which is Lyapunov stable. The function V (y) is a Lyapunov function of the dynamical system called potential. Proof: In order to show that y = 0 is an equilibrium solution we note that if condition 1. and 2. are satisfied then y = 0 is a minimum of V (y), thus ∂V ∂V = 0, = 0. (5.7) ∂y1 y=0 ∂y2 y=0 5.2 Lyapunov function method for investigating stability 7 Hence y = 0 is an equilibrium solution of the gradient flow. The function V (y) is a Lyapunov function for the gradient flow.Indeed it satisfies conditions 1. and 2. and its the orbital derivative is non-negative, as " 2  2 # ∂V ∂V ∂V ∂V Df (V ) = ẏ1 + ẏ2 = − + ≤ 0. (5.8) ∂y1 ∂y2 ∂y1 ∂y2 From the Lyapunov stability theorem it follows that the equilibrium solution y = 0 is Lyapunov stable. Example:Verify that the following dynamical system is a gradient flow and determine its Lyapunov function (potential): ẏ1 = −y1 − y1 y22 , ẏ2 = −2y2 − y12 y2 (5.9) We want to express the dynamical system as a gradient flow of V (y1 , y2 ), i.e. ∂V f1 (y1 , y2 ) = −y1 − y1 y22 = − (5.10) ∂y1 ∂V f2 (y1 , y2 ) = −2y2 − y12 y2 = −. (5.11) ∂y2 Integrating the first equation we get Z 1 1 V (y1 , y2 ) = − f1 (y1 , y2 )dy1 + g(y2 ) = y12 + y12 y22 + g(y2 ). (5.12) 2 2 Imposing f2 (y1 , y2 ) = −2y2 −y12 y2 = − ∂y∂V 2 we obtain g 0 (y2 ) = −2y2 leading to g(y2 ) = y22 +C. Imposing V (0) = 0 we get 1 1 V (y1 , y2 ) = y12 + y22 + y12 y22 (5.13) 2 2 It is easy to check that V (y) > 0 for y 6= 0. Therefore V (y1 , y2 ) conditions 1. and 2. and is the Lyapunov function (potential function) of the considered dynamical system. Moreover, the considered dynamical system is the gradient flow of V (y) and its equilibrium solution y = 0 is Lyapunov stable. The Lyapunov function method enables to investigate the stability of whole classes of sys- tems of ODEs. For example, along these lines one can prove the following important gener- alization of the theorem on p.4, which we state without proof: Theorem: Let us consider a nonlinear system of two ODEs of the form     y˙1 y1 =A + higher order nonlinear terms , (5.14) y˙2 y2 where the matrix A is time independent and characterized by the two eigenvalues λ1 , λ2. Then 8 5 Stability of Solutions of ODEs 1. if both Reλ1 < 0 and Reλ2 < 0 then the zero solution of (5.14) is asymptotically stable. 2. If at least one of Reλ1 , Reλ2 is positive then the zero solution of (5.14) is unstable. 3. If max{Reλ1 , Reλ2 } = 0 then the stability of the zero solution is determined not only by A but also by the properties of the nonlinear terms, i.e., the zero solution may be stable for some nonlinear terms but unstable for others. Note: The third case implies that linear stability analysis does not work if max{Reλ1 , Reλ2 } = 0. Example: Determine the maximal range of the values of the parameter a for which the zero solution of the system y˙1 = y1 + (2 − a)y2 , y˙2 = ay1 − 3y2 + (a2 − 2a − 3)y12 is (i) unstable, (ii) stable. Solution: The linear part of the system is obtained by simply discarding the nonlinear terms in the second  equation,  as can be verified by Taylor expansion. Hence it is described by the matrix 1 2−a A= whose characteristic equation is λ2 + 2λ + (a2 − 2a − 3) = 0. The two a −3 roots are given by √ √ λ1 = −1 + −a2 + 2a + 4 , λ2 = −1 − −a2 + 2a + 4. If the two roots are complex conjugate we have Re{λ1,2 } = −1 < 0, hence the zero solution is asymptotically stable. We conclude that an instability √ may occur only for values of a where both roots are real and λ1 > 0. This implies −a + 2a + 4 > 1 so that −a2 + 2a + 4 > 1 2 or equivalently a2 − 2a − 3 = (a + 1)(a − 3) < 0 , hence −1 < a < 3. Thus for a ∈ (−1, 3) the zero solution is unstable. Correspondingly, for a < −1 or a > 3 the zero solution must be asymptotically stable. For a = −1 or a = 3 we have λ1 = 0 while λ2 = −2 < 0. Therefore in this case the situation depends on the nonlinear terms. But precisely for these parameter values of a the nonlinear term in our original system vanishes, due to a2 − 2a − 3 = 0. The system thus becomes linear with eigenvalues λ1 = 0, λ2 = −2, hence by our theorem on page 4 of this document for a = −1, 3 the zero solution is stable but not asymptotically stable.