1.3. Stationary Points and Closed Orbits 21 by γ gives n = + r with 0 r γ. Show that the remainder, r, must be zero.) A solution σ is called periodic if it is nonconstant and has a non- trivial period, so that by the proposition all its periods are multiples of a smallest positive period γ, called the prime period of σ. A real number T is called a period of the time-dependent vector field V if V (x, t) = V (x, t + T ) for all t T and x R. A repeat of the arguments above show that the set Per(V ) of all periods of V is again a closed subgroup of R, so again there are three cases: 1) Per(V ) = R, i.e., V is time-independent, 2) Per(V ) = {0}, i.e., V is nonperiodic, or 3) there is a smallest positive element T0 of Per(V ) (the prime period of V ) and Per(V ) consists of all integer multiples of this prime period. Exercise 1–17. Show that if T is a period of the time-dependent vector field V and σ is a solution of dx dt = V (x, t), then T is also a period of σ provided there exists a real number t1 such that σ(t1) = σ(t1 + T ). (Hint: Use the uniqueness theorem.) Note the following corollary: in the autonomous case, if an orbit σ “comes back and meets itself”, i.e., if there are two distinct times t1 and t2 such that σ(t1) = σ(t2), then σ is a periodic orbit and t2 t1 is a period. For this reason, periodic solutions of autonomous ODEs are also referred to as closed orbits. Another way of stating this same fact is as follows: 1.3.2. Proposition. Let φt be the flow generated by a complete, autonomous ODE, dx dt = V (x). A necessary and sufficient condition for the maximum solution curve σp with initial condition p to be periodic with period T is that p be a fixed point of φT . Example 1–4. For the harmonic oscillator system in R2: dx dt = −y, dy dt = x, we have seen that the solution with initial condition (x0,y0) is x(t) = x0 cos(t) y0 sin(t), y(t) = x0 sin(t) + y0 cos(t). Clearly the origin is a stationary point, and every other solution is periodic with the same prime period 2π.
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