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An autonomous periodic orbit has a neutral Floquet multiplier

Statement

Let $f$ be an autonomous $C^{1}$ vector field, let $γ (t) = ϕ_{t} (x_{*})$ be a nonconstant periodic solution of period $T > 0$ , and let $M = D_{x} ϕ_{T} (x_{*})$ be its monodromy matrix. Then

M f (x_{*}) = f (x_{*}) .

In particular, $1$ is a Floquet multiplier.

Facts & Assumptions

Given: The autonomous vector field, its $C^{1}$ local flow on a common tube through time $T$ , and the nonconstant periodic solution $γ (T) = x_{*}$ .

[F1]

The variational derivative $D_{x} ϕ_{T} (x_{*})$ exists on a common compact tube (Initial-data sensitivity of an ODE flow on a common compact tube).

[F2]

A $C^{1}$ autonomous field has a unique maximal $C^{1}$ flow with $\partial_{t} ϕ_{t} (x) = f (ϕ_{t} (x))$ (C¹ Euclidean maximal flows, variational dependence and the finite C² upgrade). Uniqueness gives $ϕ_{T} (ϕ_{s} (x_{*})) = ϕ_{T + s} (x_{*})$ wherever both sides are defined: as functions of the elapsed time, both sides solve the same IVP from $ϕ_{s} (x_{*})$ .

[F3]

Derivatives of compositions obey the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F4]

A monodromy matrix is the period-time variational fundamental matrix, and its eigenvalues are the Floquet multipliers (Local periodic-orbit, Poincare-return, and Floquet data).

Proof

1.1givenF2

Nonconstancy implies $f (x_{*}) \neq 0$ : otherwise the constant curve through $x_{*}$ is an IVP solution, and uniqueness in [F2] forces $γ$ to be that constant curve.

2.1F1F2F3F4step 1.1∎

Differentiate the flow identity in [F2] with respect to $s$ at $s = 0$ . The chain rule [F3] gives $D_{x} ϕ_{T} (x_{*}) f (x_{*})$ on the left, while the time derivative on the right is $f (ϕ_{T} (x_{*})) = f (x_{*})$ . By [F1] the left derivative is the monodromy action, hence $M f (x_{*}) = f (x_{*})$ . Since $f (x_{*}) \neq 0$ by step 1.1, this is an eigenvector with eigenvalue $1$ , which is a Floquet multiplier by [F4].

Depends on

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Sources

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Chapter 1, §1.5 (standard reference, not scraped)
E. J. Doedel and B. E. Oldeman, AUTO-07P full manual (2021) (standard reference, not scraped)