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(ol<e. o( is asymptotically stable if it is stable and the convergence r(t)-co(t)l-0 is guaranteed as long as (0)-ro0)) is small enough. o( exponentially stable if, in addition, there exist o, C>0 such th x(t)-x0()≤Cexp(-at)|x(0)-xo(0)|t≥0 (9.8) whenever x(0)-zo(o)l is small enough3 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)| < δ. x0(·) is asymptotically stable if it is stable and the convergence |x(t) − x0(t)| � 0 is guaranteed as long as |x(0)−x0(0)| is small enough. x0(·) exponentially stable if, in addition, there exist �, C > 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