Periodic orbit

A periodic orbit is a nonconstant trajectory that returns to its initial state after some least positive period.

For $\dot{r}=r[1-r^2(2-\cos 2\theta )]$, $\dot{\theta }=2$, the substitution $w=r^{-2}$ gives $dw/d\theta +w=2-\cos 2\theta$. Its positive periodic solution yields the displayed orbit, with full planar period $\pi$. Every other solution at the same phase has $w-w_p=C{e}^{-2t}$, proving uniqueness and attraction. The limiting circles $r=1/\sqrt{3}$ and $r=1$ give a trapping annulus, with tangencies at their extremal angles.

A periodic orbit of a differentiable interval map is superstable when its periodic-orbit multiplier vanishes. A cycle passing through a critical point has zero multiplier because the derivative product includes $f^{\prime }(c)=0$. For the nonconstant quadratic family $f_{\mu }(x)=1-\mu x^2$, a superstable $2^n$ cycle satisfies $f_{\mu }^{2^n}(0)=0$ and does not have a smaller critical period.

// Target: dynamical-systems.bigb

A periodic orbit of a map whose least period is two. Its two points are fixed by the second iterate but are not fixed by the original map.

For $F(x,y)=(y,a(x-x^3/3)-y)$, a genuine two-cycle obeys $F(x,y)=(y,x)$ with $x\ne y$. Subtracting its two component equations gives $x^2+xy+y^2=3$, then substitution gives the displayed factorization. Cancelling $x+y$ loses the antipodal cycle $(\sqrt{3},-\sqrt{3})\leftrightarrow (-\sqrt{3},\sqrt{3})$, present for every $a>0$. The nonzero-sum family instead has $xy=3/a$ and $x+y=\pm \sqrt{3+3/a}$, requiring $a>3$. Retaining all vanishing factors is essential in polynomial cycle classification.

The nonzero-sum two-cycles of the cubic swap map exist for $a>3$. Their two-step characteristic polynomial has trace $T$ and determinant $D$ as displayed. Since $1-T+D=-2(a-3)(a+1)<0$, one periodic-orbit multiplier exceeds one, proving instability throughout. The other crosses minus one at $a=(1+\sqrt{29})/2$; the cycles are saddles before that threshold and repelling afterwards. At $a=3$ they meet the fixed branches in an unstable subcritical-flip branch.

Along the antipodal two-cycle of $F(x,y)=(y,a(x-x^3/3)-y)$, the Jacobian matrix is $B=\left(\begin{array}{cc}0& 1\\ -2a& -1\end{array}\right)$ at both points. Its two-step periodic-orbit multipliers are squares of the roots of ${\lambda }^2+\lambda +2a=0$. The Jury criterion makes the cycle attracting precisely in the displayed interval, and repelling for $a>1/2$. This stable family is distinct from the always unstable nonzero-sum cycles.

A limit cycle is an isolated periodic orbit of a dynamical system. It is attracting if nearby orbits approach it in forward time. A continuous family of closed orbits around a center equilibrium does not consist of limit cycles, because its members are not isolated.

Near a hyperbolic attracting limit cycle, a weak perturbation can be projected onto the neutral phase direction while transverse perturbations decay. Solving the periodic adjoint variational equation gives a phase response function. Averaging its inner product with weak forcing or coupling yields the first-order phase dynamics.

Linearizing weakly coupled locked phases gives $0=\xi -\Omega 1-kL\theta$, where $L$ is the directed graph Laplacian. Multiplying by a normalized left null vector ${\pi }^{\mathsf{T}}$ eliminates coupling and gives the displayed weighted frequency. In a balanced network, including symmetric reciprocal coupling, $\pi =1/N$ and the frequency is the arithmetic mean. A zero diagonal in the coupling matrix alone does not imply balance. For two directed weights $p,q>0$, the weights are $(q,p)/(p+q)$.

A periodic phase response function describes the first-order phase shift produced by a perturbation along a limit cycle. With period-one orbit $U$, choose the periodic adjoint solution normalized by ${\int }_0^{1}Y\cdot U^{\prime }ds=1$. The Fredholm solvability condition then projects forcing $p$ to ${\int }_0^{1}Y\cdot pds$. For voltage-difference coupling, the averaged interaction is $h(\varphi )={\int }_0^1{Y}^1(s)U^1(s+\varphi )ds$.

For an autonomous ordinary differential equation, the tangent to a periodic orbit gives a unit Floquet multiplier. A hyperbolic attracting cycle has that multiplier simple and all transverse multipliers strictly inside the unit disk. Thus the periodic variational operator has a one-dimensional null space spanned by the tangent. Nonlinear asymptotic attraction without hyperbolicity does not suffice: $\dot{r}=-(r-1{)}^3$, $\dot{\phi }=2\pi$ attracts to a unit-period circle but its radial variational equation is neutral.

The planar system $\dot{x}=y+(1-x^2-y^2)x$, $\dot{y}=-x+(1-x^2-y^2)y$ has polar coordinates $\dot{r}=r(1-r^2)$, $\dot{\theta }=-1$. Every nonzero orbit approaches the clockwise unit-circle limit cycle, of period $2\pi$. For $r(t_0)=r_0>0$,Interior orbits approach the origin backward in time, but exterior orbits have backward finite-time blow-up of an ordinary differential equation at $t=t_0+\frac{1}{2}\log (1-r_0^{-2})$. Thus forward convergence to a limit cycle does not imply existence for all negative times.

On the punctured plane, the vector field $\dot{x}=(1/r-1)x-y$, $\dot{y}=(1/r-1)y+x$ has the displayed polar equations. Its exact trajectories satisfy $r(t)=1+(r(0)-1)e^{-t}$ and $\theta (t)=\theta (0)+t$. Every forward trajectory approaches the unit circle, rotating counterclockwise, from inside or outside according to its initial radius. The circle is an attracting limit cycle with least period $2\pi$. The origin is not an equilibrium: the original vector field is undefined there. The singularity prevents an application of a whole-plane Bendixson-Dulac criterion to this periodic orbit.

If one coordinate of every nonstationary trajectory is strictly monotone, the system has no periodic orbit: a periodic coordinate must return to its initial value. Any trajectory on which that coordinate is constant must be checked separately.