# Floquet theory

Floquet theory is a branch of the theory of ordinary differential equations relating to the class of solutions to periodic linear differential equations of the form

${\displaystyle {\dot {x}}=A(t)x,}$

with ${\displaystyle x\in {R^{n}}}$ and ${\displaystyle \displaystyle A(t)\in {R^{n\times n}}}$ being a piecewise continuous periodic function with period ${\displaystyle T}$ and defines the state of the stability of solutions.

The main theorem of Floquet theory, Floquet's theorem, due to Gaston Floquet (1883), gives a canonical form for each fundamental matrix solution of this common linear system. It gives a coordinate change ${\displaystyle \displaystyle y=Q^{-1}(t)x}$ with ${\displaystyle \displaystyle Q(t+2T)=Q(t)}$ that transforms the periodic system to a traditional linear system with constant, real coefficients.

When applied to physical systems with periodic potentials, such as crystals in condensed matter physics, the result is known as Bloch's theorem.

Note that the solutions of the linear differential equation form a vector space. A matrix ${\displaystyle \phi \,(t)}$ is called a fundamental matrix solution if the columns form a basis of the solution set. A matrix ${\displaystyle \Phi (t)}$ is called a principal fundamental matrix solution if all columns are linearly independent solutions and there exists ${\displaystyle t_{0}}$ such that ${\displaystyle \Phi (t_{0})}$ is the identity. A principal fundamental matrix can be constructed from a fundamental matrix using ${\displaystyle \Phi (t)=\phi \,(t){\phi \,}^{-1}(t_{0})}$. The solution of the linear differential equation with the initial condition ${\displaystyle x(0)=x_{0}}$ is ${\displaystyle x(t)=\phi \,(t){\phi \,}^{-1}(0)x_{0}}$ where ${\displaystyle \phi \,(t)}$ is any fundamental matrix solution.

## Floquet's theorem

Let ${\displaystyle {\dot {x}}=A(t)x}$ be a linear first order differential equation, where ${\displaystyle x(t)}$ is a column vector of length ${\displaystyle n}$ and ${\displaystyle A(t)}$ an ${\displaystyle n\times n}$ periodic matrix with period ${\displaystyle T}$ (that is ${\displaystyle A(t+T)=A(t)}$ for all real values of ${\displaystyle t}$). Let ${\displaystyle \phi \,(t)}$ be a fundamental matrix solution of this differential equation. Then, for all ${\displaystyle t\in \mathbb {R} }$,

${\displaystyle \phi (t+T)=\phi (t)\phi ^{-1}(0)\phi (T).}$

Here

${\displaystyle \phi ^{-1}(0)\phi (T)}$

is known as the monodromy matrix. In addition, for each matrix ${\displaystyle B}$ (possibly complex) such that

${\displaystyle e^{TB}=\phi ^{-1}(0)\phi (T),}$

there is a periodic (period ${\displaystyle T}$) matrix function ${\displaystyle t\mapsto P(t)}$ such that

${\displaystyle \phi (t)=P(t)e^{tB}{\text{ for all }}t\in \mathbb {R} .}$

Also, there is a real matrix ${\displaystyle R}$ and a real periodic (period-${\displaystyle 2T}$) matrix function ${\displaystyle t\mapsto Q(t)}$ such that

${\displaystyle \phi (t)=Q(t)e^{tR}{\text{ for all }}t\in \mathbb {R} .}$

In the above ${\displaystyle B}$, ${\displaystyle P}$, ${\displaystyle Q}$ and ${\displaystyle R}$ are ${\displaystyle n\times n}$ matrices.

## Consequences and applications

This mapping ${\displaystyle \phi \,(t)=Q(t)e^{tR}}$ gives rise to a time-dependent change of coordinates (${\displaystyle y=Q^{-1}(t)x}$), under which our original system becomes a linear system with real constant coefficients ${\displaystyle {\dot {y}}=Ry}$. Since ${\displaystyle Q(t)}$ is continuous and periodic it must be bounded. Thus the stability of the zero solution for ${\displaystyle y(t)}$ and ${\displaystyle x(t)}$ is determined by the eigenvalues of ${\displaystyle R}$.

The representation ${\displaystyle \phi \,(t)=P(t)e^{tB}}$ is called a Floquet normal form for the fundamental matrix ${\displaystyle \phi \,(t)}$.

The eigenvalues of ${\displaystyle e^{TB}}$ are called the characteristic multipliers of the system. They are also the eigenvalues of the (linear) Poincaré maps ${\displaystyle x(t)\to x(t+T)}$. A Floquet exponent (sometimes called a characteristic exponent), is a complex ${\displaystyle \mu }$ such that ${\displaystyle e^{\mu T}}$ is a characteristic multiplier of the system. Notice that Floquet exponents are not unique, since ${\displaystyle e^{(\mu +{\frac {2\pi ik}{T}})T}=e^{\mu T}}$, where ${\displaystyle k}$ is an integer. The real parts of the Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if the Lyapunov exponents are nonpositive and unstable otherwise.

## References

• C. Chicone. Ordinary Differential Equations with Applications. Springer-Verlag, New York 1999.
• M.S.P. Eastham, "The Spectral Theory of Periodic Differential Equations", Texts in Mathematics, Scottish Academic Press, Edinburgh, 1973. ISBN 978-0-7011-1936-2.
• Ekeland, Ivar (1990). "One". Convexity methods in Hamiltonian mechanics. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247. ISBN 3-540-50613-6. MR 1051888.
• Floquet, Gaston (1883), "Sur les équations différentielles linéaires à coefficients périodiques" (PDF), Annales Scientifiques de l'École Normale Supérieure, 12: 47–88, doi:10.24033/asens.220
• Krasnosel'skii, M.A. (1968), The Operator of Translation along the Trajectories of Differential Equations, Providence: American Mathematical Society, Translation of Mathematical Monographs, 19, 294p.
• W. Magnus, S. Winkler. Hill's Equation, Dover-Phoenix Editions, ISBN 0-486-49565-5.
• N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964.
• Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0.
• Deng, Chunqing; Shen, Feiruo; Ashhab, Sahel; Lupascu, Adrian (2016-09-27). "Dynamics of a two-level system under strong driving: Quantum-gate optimization based on Floquet theory". Physical Review A. 94 (3). arXiv:1605.08826. doi:10.1103/PhysRevA.94.032323. ISSN 2469-9926.
• Huang, Ziwen; Mundada, Pranav S.; Gyenis, András; Schuster, David I.; Houck, Andrew A.; Koch, Jens (2021-03-22). "Engineering Dynamical Sweet Spots to Protect Qubits from 1 / f Noise". Physical Review Applied. 15 (3). arXiv:2004.12458. doi:10.1103/PhysRevApplied.15.034065. ISSN 2331-7019.
• Nguyen, L.B.; Kim, Y.; Hashim, A.; Goss, N.; Marinelli, B.; Bhandari, B.; Das, D.; Naik, R.K.; Kreikebaum, J.M.; Jordan, A.; Santiago, D.I.; Siddiqi, I. (16 January 2024). "Programmable Heisenberg interactions between Floquet qubits". Nature Physics. 20 (1): 240–246. arXiv:2211.10383. Bibcode:2024NatPh..20..240N. doi:10.1038/s41567-023-02326-7.