= Periodic solution
{title2=$x(t+T)=x(t)$}
A <periodic solution> is a <solution of a differential equation> which is a <periodic function>: for some $T>0$, $x(t+T)=x(t)$ at every time. This demands repetition of the state, rather than merely repetition of the coefficients in the <differential equation>. A nonconstant <periodic solution> of an <autonomous differential equation> traces a <periodic orbit>; constant <equilibrium points> are also <periodic functions>, so they must be excluded when asserting absence of <periodic orbits>.
For an explicit example, consider the <initial value problem>
$$
y'=(a+b\cos t)y,\qquad y(0)=1,\qquad a,b\in\mathbb R.
$$
Its coefficient has period $T=2\pi$. Taking a <logarithmic derivative> and then applying <integration> gives
$$
\log y(t)=\int_0^t(a+b\cos s)\,ds=at+b\sin t,\qquad y(t)=e^{at+b\sin t}.
$$
Direct <differentiation> verifies both the <differential equation> and the <initial condition>. After one coefficient period,
$$
y(t+2\pi)=e^{2\pi a}y(t).
$$
Thus the repeating coefficient generally produces a repeating shape with a changed amplitude. With $b=1$, the three cases $a=-0.2,0,+0.2$ have one-period factors approximately $0.2846,1,3.5136$, respectively. Only $a=0$ gives a <periodic solution> with this <initial condition>. For $a<0$ the positive <solution of a differential equation> tends to zero, and for $a>0$ it tends to infinity; neither can have any positive period. The zero <solution of a differential equation>, obtained with a different <initial condition>, is periodic for every $a$.
\Image[/periodic-coefficient-solutions-and-envelopes.png]
{title=One repeating coefficient can produce decay, periodic motion, or growth}
{description=The original example $y'=\left(a+\cos t\right)y$, $y(0)=1$, shown over one coefficient period and four coefficient periods. Shading marks the bounds $e^{at-1}\leq y(t)\leq e^{at+1}$, and dashed curves mark $e^{at}$. The right column uses a logarithmic vertical scale.}
The <exponential function> separates two roles:
$$
y(t)=\underbrace{e^{b\sin t}}_{\text{periodic modulation}}\underbrace{e^{at}}_{\text{exponential trend}},\qquad e^{at-|b|}\leq y(t)\leq e^{at+|b|}.
$$
For $b=1$, even the decaying case $a=-0.2$ initially rises, because $y'(0)=0.8>0$. The growing case $a=0.2$ has intervals of decline, because $a+\cos t$ becomes negative. This is the distinction between the instantaneous <logarithmic derivative> $y'/y$ and the <Floquet growth rate> measured over many periods.
More generally, for a real continuous <periodic function> $q$ of period $T$, the scalar <linear differential equation> $y'=q(t)y$ has
$$
y(t)=y(0)\exp\left(\int_0^t q(s)\,ds\right),\qquad m=\exp\left(\int_0^T q(s)\,ds\right).
$$
A nonzero <solution of a differential equation> is $T$-periodic exactly when $m=1$, or equivalently when the coefficient has zero integral over a period. This scalar calculation is the simplest example of a <Floquet multiplier>.<Ordinary Differential Equations and Dynamical Systems by Gerald Teschl>[§3.6, printed p. 92, scalar example following Theorem 3.15]{ref}
Back to article page
