A Generalized Averaging Theory
2.3 Averaging via Floquet Theory
2.3.2 Nonlinear Floquet Theory
By using the exponential representation of the flow of nonlinear systems, linear Floquet theory nicely ex- tends to the nonlinear time-periodic case, Equation (2.23). While hinted at in prior work [144], the extension of Floquet theory to nonlinear systems is fully developed in this section.
Theorem 8 (Nonlinear Floquet Theorem) Let ΦX0,t be the flow generated by the time-periodic differen- tial equation (2.23). If the monodromy map has a logarithm, then the flowΦX0,t can be represented as a composition of flowsΦX0,t=P(t)◦exp(Zt), whereP isT-periodic, andZis an autonomous vector field.
proof
The proof exactly follows the proof of the linear Floquet theorem [51]. It is assumed that the T-periodic vector field in (2.23) determines a flow, therefore the time shifted version,τ =t+T, does so also. The flow of the time-shifted version is the solution to the differential equation,
dx
dτ =X(x, τ).
The flowsΦX0,tandΦX0,τ differ by an invertible mapping,Ψ,
ΦX0,τ = ΦX0,t+T = ΦX0,t◦Ψ.
Assume, for now, that there exists an autonomous flow denoted byΦZ0,tequalingΨat timeT. Consider, P(t)≡ΦX0,t◦ ΦZ0,t−1
. (2.27)
This defined flow,P(t), isT-periodic.
P(t+T) = ΦX0,t+T ◦ ΦZ0,t+T−1
= ΦX0,t◦Ψ◦ ΦZ0,t◦ΦZ0,T−1
= ΦX0,t◦Ψ◦ ΦZ0,T−1
◦ ΦZ0,t−1
= ΦX0,t◦ ΦZ0,t−1
=P(t).
TheT-periodicity of the original vector fieldX ensures that ΦX0,t+T = ΦX0,t◦ΦX0,T, implying that
ΦZ0,T = Ψ≡ΦX0,T, (2.28)
i.e.,Ψis the monodromy map of the flow. If we rewrite the respective flows in the chronological calculus formalism, then
exp (ZT) =−→exp Z T
0
Xτdτ
. Inverting via the logarithm,
Z = 1 T ln−→exp
Z T 0
X(x, θ) dθ
. (2.29)
This is precisely the average of theT-periodic vector field; a connection that will be made more explicit in the sequels. Conditions under which this logarithm exists are given in [2]. For now, the theorem statement assumes the existence of the logarithm, and by extension the autonomous flow assumed in the proof of the theorem.5
The mappingP(t)is called the Floquet mapping, and the vector fieldZis called the autonomous averaged vector field corresponding toX. When the vector fieldX is known through context, thenZ will simply be called the autonomous averaged vector field.
The monodromy map, Ψ, plays an important role in nonlinear Floquet theory, just as it did in linear Floquet theory. The proposition below relates the stability of the monodromy map to the stability of the original system.
Theorem 9 If the monodromy map,Ψof the system (2.23) has a fixed point, then the actual flow,ΦX0,t, has a periodic orbit whose stability is determined by the stability of the monodromy map.
proof
Suppose that the monodromy map has a fixed point denotedx∗, then the actual flow periodically returns to this fixed point, as can be seen by the following equalities:
ΦX0,0(x∗) = (P(0)◦exp(Z0)) (x∗) =x∗, and
ΦX0,T(x∗) = (P(T)◦exp(Z T)) (x∗) = (P(0)◦Ψ) (x∗) =x∗,
implying thatx∗ lies on a periodic solution to the flow,ΦX0,t, of the system (2.23). The orbit throughx∗ is the orbit of the theorem statement. The monodromy map is the Poincar´e map of (2.23), evaluated on the orbit throughx∗ [162].
If the monodromy map is asymptotically (exponentially) stable, then the correspoding orbit is asymptotically (exponentially) stable.
Corollary 2 If the flow,ΦX0,t, of system (2.23) has a fixed pointx∗, as does the monodromy map,Ψ, then stability of the fixed point under the flowΦX0,tcan be determined from the stability of the monodromy map.
In particular an asymptotically (exponentially) stable fixed point for the monodromy map implies an asymp- totically (exponentially) stable fixed point for the actual system (2.23).
5In Section 2.2 we restricted the vector fields to be smooth. Smoothness implies existence of the logarithm for finite time.
proof
To see that stability of the monodromy map implies stability of the actual flow, it will be necessary to appeal to notions of D-stability and C-stability in per Pars [131]. C-stability is the traditional notion of stability; the
“C” stands for continuous time. The “D” in D-stability refers to discrete time and involves taking discrete samples of the system response at fixed intervals. Pars shows that the two notions are equivalent when stability of a fixed point is sought. If the monodromy map is stable, then that implies D-stability for the actual system. Since D-stability and C-stability are equivalent under the conditions of interest, the system is stable. The same holds for D-instability.
Explicitly calculating the monodromy map may be problematic if not impossible. Equivalent to stability of the monodromy map is stability of the autonomous vector fieldZ whose flow gives the monodromy map.
Corollary 3 [144] The stability properties of the logarithm of the monodromy map may be used to infer the stability properties of the monodromy map itself.
Comment. When applied to the simplified case of linear stability analysis, the above conclusions lead to the following well known fact for Floquet theory: calculation of the Floquet multipliers is equivalent to calculation of the Floquet exponents. In the nonlinear theory, the monodromy map is a nonlinear transfor- mation of state, therefore it’s a bit more complicated to determine such a property. In this case, the logarithm provides a useful means to analyze stability.
Non-uniqueness of the Monodromy Map. Given two fundamental matrix solutions, ΦX0,t1 and ΦX0,t2, to (2.24) there must exist a transformation of state,Θ, fromX1toX2, i.e.,X2= Θ∗X2. Given the monodromy matrix forX1, calledΨ1, the monodromy matrix forX2isΨ2= Θ−1Ψ1Θ. As in the linear case the freedom occurs because of the ability to shift time byT.
An additional freedom may occur because of the dependence of the mondramy map calculation on initial conditions. Although the monodromy map was defined to beΨ = ΦX0,T, any choice of initial time may be used,Ψ = Φe Xt
0,t0+T, so long as there is flow for one period of time. The two monodromy maps will differ by a transformation of stateΘ, i.e.,Ψ = Θ−1ΨΘ, corresponding to flow by the difference in initial times.e
It is not known a priori whether or not a given initial condition will provide the best calculation of the monodromy map or its logarithm, therefore one should not expect for the evolution of the autonomous vector field,Z, to generically result in the best average. It may be possible for the time-periodic mapping, P, to contain a bias that leads to a better representation of the average. Suppose that
P(t) =Pe(t)P0, (2.30)
withP0 a time-independent transformation. It is possible to recover a different averaged vector field from this knowledge.
Theorem 10 Suppose the the Floquet mapping has a time-independent bias, i.e., Equation (2.30) holds.
Then the new averaged vector field is
Ze= (P0)∗Z.
proof
The proof is rather straightforward and is basically a change of coordinates. The autonomous flow is z(t) = exp(Zt)x(0)
and the actual flow is
x(t) =P(t, z(t)).
However, with the bias mapping, we obtain
x(t) =Pe(t)◦P0(z(t)).
Define the new variable,ez(t),
e
z(t) =P0(z(t)).
The evolution ofz(t)e obeys the differential equation, e˙
z=T P0 Z(P0−1(z))
= (P0)∗Y(z), ez0 =P0(x0) and the solution evolves according to
x(t) =Pe(t)◦exp(Zt)e ◦P0(x0).
The role played byP0 is equivalent toΘin the discussion of the non-uniqueness of the monodromy map.
The flexibility inherent to the decompositionP andexp(Zt)is well known in classicalnth-order averaging theory [42]. The opposite may also be desirable: to extract unecessary coordinate transformations from the averaged vector field, the reverse procedure may be performed. This is important when one would like to utilize the stability theorems of Section 2.3, because the bias might prevent the fixed point of the autonomous vector field from corresponding to the fixed point of the actual vector field.