Floquet exponents

This is an eigenvalue equation for the Floquet exponents and the coefficients To see this more clearly, let us introduce the basis of states an), where the Greek letter labels the molecular states and the Roman letter the Fourier components. Then, Eq. (8.19) can be written as... 

The size of the matrix as it operates on the perturbation vector is directly related to the eigenvalues of J (or of B). The eigenvalues of J are known as the Floquet multipliers fit the eigenvalues of B are the Floquet exponents / ,. In general the former are easier to evaluate, although we should identify the parameter p2 introduced in chapter 5 with the Hopf bifurcation formula as a Floquet exponent for the emerging limit cycle (then P2 < 0 implies stability, P2 > 0 gives instability, and P2 = 0 corresponds to a bifurcation between these two cases). 

Although most of the results can be established with mathematical rigor, there are some elusive problems. These center around the possibility of multiple limit cycles and the difficulty of determining the stability of such limit cycles. At this point one must simply make a hypothesis and resort to numerical evidence in any specific case. Determining the number of limit cycles is a deep mathematical problem, and even in very simple cases the solution is not known. Hilbert s famous sixteenth problem, concerning the number of limit cycles of a second-order system with polynomial right-hand sides, remains basically unresolved. In principle, the stability of a limit cycle can be determined from the Floquet exponents (see Section 4), but this is a notoriously difficult computation - indeed, generally an impossible one. 

As customary, the exponential of a matrix means the sum of the matrix series corresponding to the exponential function. The eigenvalues of i T) =e are called the Floquet multipliers. The eigenvalues of B are called the Floquet exponents. (There is some delicacy about the uniqueness of B which we will ignore because it is not relevant to our use.) Usually it is not possible to compute the Floquet exponents or multipliers. However, for low-dimensional systems of the kind we will investigate, there is a general theorem about the determinant of a fundamental matrix which is helpful. Let 4>(0 be a fundamental matrix for (4.1) with i (0) = I. Then... 

This system is periodic and therefore the Floquet theory described in Section 4, Chapter 3, applies. Let 4>(/) be the fundamental matrix solution of (2.2). The Floquet multipliers of (2.2) are the eigenvalues of 4>(w) if /i is a Floquet multiplier and /i = e" then A is called a Floquet exponent. Only the real part of a Floquet exponent is uniquely defined. 

In terms of Floquet exponents, the condition for stability is 9J(A) < 0 for all exponents and the condition for instability is that 9f(A) > 0 for some exponent A. Here, 9i(A) denotes the real part of A. 

It is easy to see that the numbers (yj(l) -1), i = 1,2, are the Floquet exponents corresponding to the identically zero periodic solution of (3.2). Consequently, the solution is asymptotically stable when both exponents are negative and unstable when one of the exponents is positive. Proposition 3.1 says more than this it states that competitors, is washed out of the chemostat if/(I) < 1, but this outcome has nothing to do with competition since it occurs even in the absence of the other competitor. As our main interest is in the effects of competition, we assume hereafter that... 

The outcome of competition between two competitors depends on the stability properties of the single-competitor periodic solutions Ei and p2-It turns out that the stability of these solutions is determined, in each case, by a single Floquet exponent in a biologically intuitive way. Suppose that a chemostat is charged at / = 0 with only the competitor X[. According to Proposition 3.2, the concentration Xi very rapidly approaches the level... 

Proposition 3.3. Floquet exponents of the periodic solution E are given by... 

It must be stressed that the hypotheses of Corollary 5.2 give sufficient, but not necessary, conditions for the existence of a positive periodic solution possessing strong stability properties. Furthermore, since the singlepopulation periodic solutions Eft) and 2(0 are not explicitly computable, as the corresponding rest points were in Chapter 1, it does not seem possible to obtain explicit formulas for A,2and A21. However, these crucial Floquet exponents can be easily approximated numerically. One must... 

Since the limit cycle is invariant with respect to a phase shift, one of the Floquet exponents is zero and the asymptotic stability of the limit cycle implies that the other Floquet exponents have all negative real parts. Thus the amplitude of all modes decay in time and the uniform oscillatory state is stable to small localized perturbations (within the filament model and in the linear regime to which Eq. (8.6) applies). The slowest decaying mode (i.e. for p = 0 and k = 0) decays on average as exp(—At), that is the same as the decay rate of a passive scalar inhomogeneity in the filament model. 

Spectral analysis of the linearized semiflow along a periodic solution is called Floquet theory. The eigenvalues p of the linearized period map are called Floquet multipliers. A Floquet exponent is a complex / such that exp(/3r) is a Floquet multiplier of the system, where t denotes the minimal period. A periodic solution is hyperbolic if, and only if, it possesses only the trivial Floquet multiplier p = 1 on the unit circle, and this multiplier has algebraic multiplicity one. Otherwise it is called non-hyperbolic. In ODEs hyperbolic periodic solutions possess stable and unstable manifolds, similarly to the case of hyperbolic equilibria. Non-hyperbolic periodic solutions possess center manifolds. 

W. Just, E. Reibold, K. Kacperski, P. Fronczak, J. A. Holyst, and H. Benner Influence of stable floquet exponents on time-delayed feedback control, Phys. Rev. E 61, 5045 (2000). 

A Lyapunov exponent is a generalized measure trf the growth or decay of small perturbations away from a particular dynamical state. For perturbations around a fixed point or steady state, the Lyapunov exponents are identical to the stability eigenvalues of the Jacobian matrix discussed in an earlier section. For a limit cycle, the Lyapunov exponents are called Floquet exponents and are determined by carrying out a stability analysis in which perturbations are applied to the asymptotic, periodic state that characterizes the limit cycle. For chaotic states, at least one of the Lyapunov exponents will mm out to be positive. Algorithms for the calculation of Lyapunov exponents are discussed in a later section in conjunction with the analysis of experimental data. These algorithms can be used for simulations that yield possibly chaotic results as well as for the analysis of experimental data. 

As described in a previous section, the Lyapunov exponents are a generalized measure of the growth or decay of perturbations that might be applied to a given dynamical state they are identical to the stability eigenvalues for a steady state and the Floquet exponents for a limit cycle. For aperiodic motion at least one of the Lyapunov exponents will be positive, so it is generally sufficient to calculate just the largest Lyapunov exponent. 

Remark 11.2 The eigenvalues Pj of M are known as Floquet multipliers or characteristic multipliers and the eigenvalues v, of B are known as the Floquet exponents or characteristic exponents. They are related by pi = exp(n,T) [435]. 

Imi I 1 (respectively all Floquet exponents satisfy Re n, < 0) and for all Floquet multipliers with = 1 (respectively all Floquet exponents with Rev, = 0) the algebraic and geometric multiplicities are equal. 

A periodic linear system is asymptotically stable if all Floquet multipliers satisfy )rij I < 1 (respectively all Floquet exponents satisfy Re v, < 0) [435]. 

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