Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j(Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 9: Local Behavior Near Trajectories This lecture presents results which describe local behavior of ODE models in a neigbor- hood of a given trajectory, with main attention paid to local stability of periodic solutions 9.1 Smooth Dependence on Parameters In this section we consider an ODe model i(t=a(a(t), t, u),. (to)=To(a), (9.1) here u is a parameter. When a and To are differentiable with respect to u, the solution c(t)=a(t, u)is differentiable with respect to u as well. Moreover, the derivative of r(t, u) with respect to u can be found by solving linear ODE with time-varying coefficients Theorem9.1Leta:Rn×R×R→ r be a continuous function,p∈R.Let o: to, ti b+ r be a solution of(9.1)with u= Ho. Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that (o(t), t, Ho)E X for allt e [ to, t1. Then for all u in a neigborhood of po the ODE in(9.1) has a unique solution a(t)=r(t, u. This solution is a continuously differentiable function of u, and its derivative with respect to u at u= Ho equals A(t), where 4: to, t HR is the n-by-k matric-valued solution of the ODE △(t)=A(t)△(t)+B(t),△(to)=△o, (9 where A(t) is the derivative of the mapiHa(i, t, uo) with respect to i atI=ro(t), B(t) is the derivative of the map u Ha(ao(t),t, u)at A=Ho, and Ao is the derivative of the mp口o(1)atu=p Version of october 10. 2003
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.243j (Fall 2003): DYNAMICS OF NONLINEAR SYSTEMS by A. Megretski Lecture 9: Local Behavior Near Trajectories1 This lecture presents results which describe local behavior of ODE models in a neigborhood of a given trajectory, with main attention paid to local stability of periodic solutions. 9.1 Smooth Dependence on Parameters In this section we consider an ODE model x˙ (t) = a(x(t), t, µ), x(t0) = ¯x0(µ), (9.1) where µ is a parameter. When a and x¯0 are differentiable with respect to µ, the solution x(t) = x(t, µ) is differentiable with respect to µ as well. Moreover, the derivative of x(t, µ) with respect to µ can be found by solving linear ODE with time-varying coefficients. Theorem 9.1 Let a : Rn × R × Rk ∞� Rn be a continuous function, µ0 ≤ Rk. Let x0 : [t0, t1] ∞� Rn be a solution of (9.1) with µ = µ0. Assume that a is continuously differentiable with respect to its first and third arguments on an open set X such that (x0(t), t, µ0) ≤ X for all t ≤ [t0, t1]. Then for all µ in a neigborhood of µ0 the ODE in (9.1) has a unique solution x(t) = x(t, µ). This solution is a continuously differentiable function of µ, and its derivative with respect to µ at µ = µ0 equals �(t), where � : [t0, t1] ∞� Rn,k is the n-by-k matrix-valued solution of the ODE �(˙ t) = A(t)�(t) + B(t), �(t0) = �0, (9.2) where A(t) is the derivative of the map x¯ ∞� a(¯x, t, µ0) with respect to x¯ at x¯ = x0(t), B(t) is the derivative of the map µ ∞� a(x0(t), t, µ) at µ = µ0, and �0 is the derivative of the map µ ∞� x¯0(µ) at µ = µ0. 1Version of October 10, 2003
Proof Existence and uniqueness of r(t, u)and A(t)follow from Theorem 3. 1. Hence, in order to prove differentiability and the formula for the derivative, it is sufficient to show that there exist a function C: R++R+ such that C(r)/r-0 as r-0 and E>0 such x(t,)-△(t)(u-10)-0(t)≤C(-ol enever u-pol E. Indeed, due to continuous differentiability of a, there exist C1,E0 such that a(x,t,)-a(xo(1),t,0)-4(t)(z-x0(t)-B(t)(-p0川≤C1(-0(t)+u-po|) o(p)-5(0)-△0(4-0)≤C1(l-o) whenever z-x0(t)+|a-A|≤e,t∈[to,t] Hence for ()=x(t2p)-r0(t)-△(t)(u-10) we nave 16(1)|≤C216(t)|+C3(|-po), as long as 8(t)l 0 is sufficiently small. Together with 16(tl)≤C4(lu-0) this implies the desired bound Example 9.1 Consider the differential equation v()=1+sin(y(t),y(0)=0, where u is a small parameter. For u=0, the equation can be solved explicitly: yo(t)=t Differentiating yu(t) with respect to u at u=0 yields A(t) satisfying △()=sin(t),△(0)=0, i.e. A(t)=1-cos(t). He ence (t)=t+(1-cos(t)+O(12) for small
2 Proof Existence and uniqueness of x(t, µ) and �(t) follow from Theorem 3.1. Hence, in order to prove differentiability and the formula for the derivative, it is sufficient to show that there exist a function C : R+ ∞� R+ such that C(r)/r � 0 as r � 0 and δ > 0 such that |x(t, µ) − �(t)(µ − µ0) − x0(t)| ≈ C(|µ − µ0|) whenever |µ − µ0| ≈ δ. Indeed, due to continuous differentiability of a, there exist C1, δ0 such that |a(¯x, t, µ) − a(x0(t), t, µ0) − A(t)(¯x − x0(t)) − B(t)(µ − µ0)| ≈ C1(|x¯ − x¯0(t)| + |µ − µ0|) and |x¯0(µ) − x¯0(µ0) − �0(µ − µ0)| ≈ C1(|µ − µ0|) whenever |x¯ − x¯0(t)| + |µ − µ0| ≈ δ, t ≤ [t0, t1]. Hence, for �(t) = x(t, µ) − x0(t) − �(t)(µ − µ0) we have |� ˙(t)| ≈ C2|�(t)| + C3(|µ − µ0|), as long as |�(t)| ≈ δ1 and |µ − µ0| ≈ δ1, where δ1 > 0 is sufficiently small. Together with |�(t0)| ≈ C4(|µ − µ0|), this implies the desired bound. Example 9.1 Consider the differential equation y˙(t) = 1 + µ sin(y(t)), y(0) = 0, where µ is a small parameter. For µ = 0, the equation can be solved explicitly: y0(t) = t. Differentiating yµ(t) with respect to µ at µ = 0 yields �(t) satisfying �(˙ t) = sin(t), �(0) = 0, i.e. �(t) = 1 − cos(t). Hence yµ(t) = t + µ(1 − cos(t)) + O(µ2 ) for small µ
9. 2 Stability of periodic solutions In the previous lecture, we were studying stability of equilibrium solutions of differential equations. In this section, stability of periodic solutions of nonlinear differential equations is considered. Our main objective is to derive an analog of the lyapunov's first method stating that a periodic solution is asymptotically stable if systems linearization around the solution is stable in a certain sense 9.2.1 Periodic solutions of time-varying ODE Consider system equations given in the form i(t)=f(a(t),t) (9.3) where f: RXRHR is continuous. Assume that a is(T, ir-periodic, in the sense that there exist T>0 ander such that ∫(t+T,r)=f(t,r),f(t,r+)=f(t,r)wt∈R,r∈R (9.4) Note that while the first equation in(9. 4)means that f is periodic in t with a period T, it is possible that i =0, in which case the second equation in(9.4)does not bring any dditional information Definition A solution To: R+ R of a(T, i)-periodic system(9.)is called (T, i r0(t+7)=x0(t)+t∈R. Example 9.2 According to the definition, the solution y(t)=t of the forced pendulum equation i(t)+i(t)+sin(y(t))=1+sin(t) (9 as a periodic one(use T =i= 27). This is reasonable, since y(t)in the pendulum equation represents an angle, so that shifting y by 2m does not change anything in the system equations. Definition A solution o: [ to, oo)HR of(9.3)is called stable if for every 8>0there exists∈>0 such that x(t)-x0()≤6Vt≥ (9.7) whenever x( is a solution of ( 9.3)such that r(0)-to(ol0 such th x(t)-x0()≤Cexp(-at)|x(0)-xo(0)|t≥0 (9.8) whenever x(0)-zo(o)l is small enough
3 9.2 Stability of periodic solutions In the previous lecture, we were studying stability of equilibrium solutions of differential equations. In this section, stability of periodic solutions of nonlinear differential equations is considered. Our main objective is to derive an analog of the Lyapunov’s first method, stating that a periodic solution is asymptotically stable if system’s linearization around the solution is stable in a certain sense. 9.2.1 Periodic solutions of time-varying ODE Consider system equations given in the form x˙ (t) = f(x(t), t), (9.3) where f : Rn ×R ∞� Rn is continuous. Assume that a is (π, xˆ)-periodic, in the sense that there exist π > 0 and xˆ ≤ Rn such that f(t + π, r) = f(t, r), f(t, r + xˆ) = f(t, r) � t ≤ R, r ≤ Rn. (9.4) Note that while the first equation in (9.4) means that f is periodic in t with a period π , it is possible that xˆ = 0, in which case the second equation in (9.4) does not bring any additional information. Definition A solution x0 : R ∞� Rn of a (π, xˆ)-periodic system (9.3) is called (π, xˆ) periodic if x0(t + π ) = x0(t) + xˆ � t ≤ R. (9.5) Example 9.2 According to the definition, the solution y(t) = t of the forced pendulum equation y¨(t) + y˙(t) + sin(y(t)) = 1 + sin(t) (9.6) as a periodic one (use π = xˆ = 2�). This is reasonable, since y(t) in the pendulum equation represents an angle, so that shifting y by 2� does not change anything in the system equations. Definition A solution x0 : [t0,→) ∞� Rn of (9.3) is called stable if for every � > 0 there exists δ > 0 such that |x(t) − x0(t)| ≈ � � t ∀ 0, (9.7) whenever x(·) is a solution of (9.3) such that |x(0) − x0(0)| 0 such that ∈x(t) − x0(t)∈ ≈ C exp(−�t)|x(0) − x0(0)| � t ∀ 0 (9.8) whenever |x(0) − x0(0)| is small enough
To derive a stability criterion for periodic solutions o: RH R of (9.3),assume continuous differentiability of function f= f(, t)with respect to the first argument D for I-to(t)l 0 is small, and differentiate the solution as a function of initial conditions r(O)a ro(O) Theorem 9. 2 Let f: R"xR+r be a continuous (T, i)-periodic function. Let o: RH R be a(T, i)-periodic solation of(9.3). Assume that there erists e>0 such that f is continuously differentiable with respect to its first argument for li-co(t0. A non-constant(T, i)-periodic solution o: RHR" of system (9. 11) is called a stable limit cycle if
4 To derive a stability criterion for periodic solutions x0 : R ∞� Rn of (9.3), assume continuous differentiability of function f = f(¯x, t) with respect to the first argument x¯ for |x¯ − x0(t)| ≈ δ, where δ > 0 is small, and differentiate the solution as a function of initial conditions x(0) � x0(0). Theorem 9.2 Let f : Rn × R ∞� Rn be a continuous (π, xˆ)-periodic function. Let x0 : R ∞� Rn be a (π, xˆ)-periodic solution of (9.3). Assume that there exists δ > 0 such that f is continuously differentiable with respect to its first argument for |x¯ − x0(t)| 0. A non-constant (π, xˆ)-periodic solution x0 : R ∞� Rn of system (9.11) is called a stable limit cycle if
(a) for every e>0 there exists 8>0 such that dist(a(t), 1o())0 such that dist(r(t), o()-0 as t-o for every solution of (9. 11) such that dist(=(0), o()0 such that a is continuously differentiable on the set X={∈R:|z-x0(t)< e for some t∈R Let A: 0,H R,n be defined by(9.9), (910).Then (a) if all eigenvalues of A(r) except one have absolute value less than 1, io( is a stable (b) if one eigenvalue of A(r has absolute value greater than 1, ro( is not a stable limit cycle
5 (a) for every δ > 0 there exists � > 0 such that dist(x(t), x0(·)) 0 such that dist(x(t), x0(·)) � 0 as t � → for every solution of (9.11) such that dist(x(0), x0(·)) 0 such that a is continuously differentiable on the set X = {x¯ ¯ ≤ Rn : |x − x0(t)| < δ for some t ≤ R. Let � : [0, π ] ∞� Rn,n be defined by (9.9),(9.10). Then (a) if all eigenvalues of �(π ) except one have absolute value less than 1, x0(·) is a stable limit cycle; (b) if one eigenvalue of �(π ) has absolute value greater than 1, x0(·) is not a stable limit cycle
