Multiplier Floquet

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

Recalling that x is identically 9SS and hence always greater than or equal to unity, we see that the Floquet multiplier is always negative with this scheme only a stable limit cycle emerges. 

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). 

An n-variable system has n associated Floquet multipliers one of these is always equal to +1. This latter point arises because if our initial perturbation is in a direction exactly along the limit cycle, then it will neither decay nor grow—there will just be a shift in phase. The remaining multipliers may be real or occur as conjugate complex pairs. The values of the multipliers can be represented by points on the complex plane, as shown in Fig. 13.17. 

Fig. 13.17. Floquet multipliers lying within the unit circle, indicating a stable periodic motion if the CFM leaves the unit circle through — 1 a period doubling occurs if it goes out through + 1 there is a saddle-node bifurcation with the disappearance of the periodic solution.

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

We now turn to an example where full use has been made of the bifurcation analysis based on Floquet multipliers, as described in 5.4.3. 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

Let us examine more closely what occurs on the right-hand boundary of the 1/1 resonance horn [Fig. 9(a)]. In a sequence of one-parameter bifurcation diagrams with respect to oj/co0, each taken at a successively higher forcing amplitude FA, we observe that as FA increases, the bifurcation point to a torus changes. The point of exit of the Floquet multipliers of the periodic... 

The differential equation for M in (7) is non-antonomous and involves evaluation of the jacobian of the forced-model equations at the current value of the trajectory jc(x0, p, t) for each time step so that it must be integrated simultaneously with the system equations. Upon convergence on a fixed point, the matrix M becomes the monodromy matrix whose eigenvalues are those of the jacobian of the stroboscopic map evaluated at the fixed point and are called the Floquet multipliers of the periodic solution. 

The Floquet multipliers determine the stability and character of the fixed point (or limit cycle) much in the same way as the eigenvalues of the jacobian... 

Critical fixed points that are undergoing bifurcation can be found by augmenting the set of fixed-point equations (5) with one of these Floquet multiplier conditions ... 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

FIGURE 4 Illustration of the three qualitatively different period doublings that occur on the segments FUE, EJ and JF. Point E has two Floquet multipliers at — I and point J is a metacritical period doubling bifurcation point. 

Among all of the points on the period 1 Hopf curve, some will have complex Floquet multipliers A with a phase angle 6 of mln)2u with n = 3 or 4 (i.e. third or fourth roots of unity) and are called hard reasonances. Because these points are fixed points for Fn that have multipliers equal to A" = 1, it is not surprising to find that subharmonic fixed points of period n are involved in addition to the bifurcating period 1 fixed point. 

FIGURE 8 (a) Detail of the tip of the 3/1 resonance horn illustrating typical way in which period 3 resonance horns close around a point with Floquet multipliers at the third root of unity (point F). (fa) and (c) The saddle-node pairings change from section AA to section BB and the unstable manifolds of the period 3 saddles no longer make up a phase locked torus. (d) A one-parameter vertical cut through the third root of unity point. The three saddles coalesce with the period one focus that is undergoing the Hopf bifurcation. 

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... 

Thus the product of the Floquet multipliers is the determinant of (7"). Equation (4.4) will be useful in some stability calculations. 

Theorem 4.2. Let n — of the Floquet multipliers of (4.6) lie inside the unit circle in the complex plane. Then 7 is an asymptotically orbitally stable trajectory of (4.5). 

Proof. The quantity under the integral sign in the definition of A in (5.2) is the trace of the Jacobian matrix for the system (5.1) evaluated along the periodic orbit. Theorem 4.2 then applies. A periodic orbit for an autonomous system has one Floquet multiplier equal to 1. Since there are only two multipliers and one of them is 1, is the remaining one. The periodic orbit is asymptotically orbitally stable because, in view of Lemma 5.1, A<0. ... 

Proof. Let 7 = (a (/), (/), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted that if there are several orbits then one must be asymptotically stable, by our assumption of hyperbolicity.) Let the Floquet multipliers of 7, viewed as a solution of (3.1), be 1 and p, where 0periodic orbit, define p( 3) by... 

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. 

Let the Floquet multipliers (eigenvalues of (7 )) of the variational equation be l,Pi,P2,. ..,p i, where the terms are listed according to multiplicity and the first one corresponds to the eigenvector e. Finally, recall from the fundamental theory of ordinary differential equations [H2, chap. 1, thm. 3.3] that... 

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