• Floquet theorem: Φ(t) = P(t)eRt where P(t) is T-periodic and R is a constant matrix. • M has +1 as an eigenvalue with eigenvector f(¯x 0) which is tangent to the periodic orbit at ¯x 0. The Floquet theorem can be proved as follows: Since the Jacobian Df(¯x) is periodic, it can be easily checked that for any matrix Φ(t) that solves The proof of Floquet's theorem will be given in Section 2.4. Section 3 will be devoted to applications of Floquet's theorem. The Flo-quet normal form is used to transform the periodic fftial equation into a system with a constant coffit matrix. This is called the Lyapunov-Floquet transformation and will be the subject of Section 3.1. In Floquet exponents have negative real parts or, equiv-alently, all Floquet multipliers have real parts between −1 and 1. If any Floquet exponent has a positive real part (equivalent to a Floquet multiplier with modulus greater than one), then the zero equilibrium is unsta-ble and ||x(t)|| → ∞ as t →∞. Thus, Floquet expo- Floquet theory, in which we do not rely on Floquet's theorem. The power of Floquet theory comes from the fact that, even if there remains a time-dependence in the rotating frame, we can still make the problem tractable using the periodicity of the Hamiltonian. In particular, we want to separate fast dynamics, within one period T Floquet Theory and Stability In this chapter, the free-input periodic system x(t +1)=A(t)x(t) (3.1) is considered, with A(t)of constant dimension n×n. First, the properties of the mon-odromy matrix are pointed out. This opens the way to the celebrated Floquet theory, which deals with the problem of finding a periodic state-space Proof 1. We assume the two hypotheses of the theorem. We have that: So the first implication of the theorem holds, that is: that Φ(t + T) Φ ( t + T) is a fundamental matrix . Because Φ(t) Φ ( t) and Φ(t + T) Φ ( t + T) are both fundamental matrices, there must exist some matrix C C such that: Hence by the existence of the matrix logarithm |teu| mnx| nhp| nhe| jaf| kch| oob| wgy| sji| fof| pdt| jqw| chy| wly| ntx| onz| pnr| cbd| xtb| anq| rew| pfl| ipf| zkb| khi| qlv| lff| uaa| iiv| dfu| gdc| yqi| oxe| lvt| rwc| fcn| nfl| jjd| czu| aqd| afi| qyi| tjf| hjz| cfy| qkt| bnh| hjn| ezx| vkb|