Poincaré maps and Floquet multipliers

Let a smooth autonomous vector field have a nonconstant periodic solution gamma of minimal period T. At a point on it, the velocity is nonzero. Choose a codimension-one local section whose normal is not perpendicular to that velocity. Transversality lets the implicit function theorem determine a smooth nearby return time and return map for points sufficiently close to the periodic point. This is a local construction near a known returning orbit, not a claim that every trajectory in the whole state space returns.

The periodic orbit intersects this chosen local section at a fixed point of the return map. If all derivative eigenvalues there have modulus strictly less than one, that fixed point attracts and the orbit is locally orbitally asymptotically stable. A multiplier outside the unit circle gives instability. Unit-modulus multipliers require nonlinear analysis, just as zero-real-part eigenvalues do for equilibria. A chosen section can have multiple global intersections per revolution; define the intended local return and orientation before inferring a period from an intersection count.

Along gamma, infinitesimal perturbations satisfy a linear equation with periodic coefficient matrix A(t), the Jacobian evaluated on the orbit. Let X solve X dot = A(t)X with X(0) equal to the identity. The matrix X(T) is the monodromy matrix; its eigenvalues are Floquet multipliers. Their moduli measure growth or decay per revolution. Floquet exponents describe growth per unit time and exponentiate over T to the multipliers, with logarithm branches relevant for complex values.

An autonomous periodic orbit always has a unit multiplier in the tangent flow direction, because a time-shifted trajectory remains a solution. The transverse return map removes that direction; its spectrum equals the remaining multipliers, counting multiplicity. For an attracting hyperbolic cycle the tangent unit multiplier is simple and every other multiplier lies strictly inside the unit circle. Requiring every full monodromy multiplier to have modulus less than one would incorrectly exclude every autonomous cycle, including stable ones.

For the polar model r dot = mu r−r cubed and theta dot = omega, with positive mu and omega, the cycle radius is square root of mu. Linear radial perturbations have exponent minus twice mu. A section at a fixed angle returns after T=2pi divided by omega, so its scalar multiplier is exp of minus twice mu times T. It is positive and less than one. A phase perturbation along the cycle, however, neither grows nor decays in the constant-frequency model.

An oriented planar-flow return map has positive scalar derivative: nearby ordered trajectories cannot cross and reverse that order during one return. Higher-dimensional sections can have more varied multiplier spectra, including negative multipliers associated with flip phenomena. Estimating return derivatives from a coarse event detector can mix transverse error with crossing-time error. Integrate the variational equation or refine the return event when quantitative stability matters. Finite-time contraction seen in a few returns also does not replace checking that the cycle and section satisfy the local theorem's assumptions.