A Generalized Averaging Theory
2.4 A General Averaging Theory
This section demonstrates how the chronological calculus, nonlinear Floquet theory, and truncations of series expansions combine to form a generalized averaging theory. Nonlinear Floquet theory successfully decomposes the flow of a time-periodic system into the composition of a time-periodic mapping and the flow of an autonomous vector field. Classical higher-order averaging theory suggests the form of the autonomous vector field and the compensatory periodic mapping. Hence, the nonlinear Floquet theorem indicates that the compensatory mapping of classical averaging theory is the Floquet mapping,P(t). Here, we demonstrate how series expansions and the chronological calculus are used to construct the compensatory (or Floquet) mapping and the autonomous averaged vector field.
First-order averaging. As a simple application, let’s revisit first-order averaging. Unlike second- and higher-order averaging, first-order averaging does not involve a compensatory mapping. This is because
Trunc0(P(t)) = Trunc0 ΦX0,t◦exp (−Zt)
= Id. (2.36)
Since the compensatory mapping is the identity, theT-periodicity constraint is trivially satisfied. This leaves the autonomous vector field,
Z = 1 T
Z T 0
Xτdτ =X, (2.37)
as the only important element in performing first-order averaging. Floquet theory can be applied to obtain the standard facts concerning stability of the average vector field flow and its relation to the actual flow.
Second-order averaging. The benefits of the chronological calculus become more apparent when we reconstruct the second-order averaging theorem of Sanders and Verhulst, c.f. Theorem 5. TruncatingP(t) results in
Trunc1(P(t)) = Id + Z t
0
Xτ−X
dτ +O 2
. (2.38)
In other words,
x(t) =z(t) + Z t
0
X(z(t), τ)−X(z(t))
dτ +O 2
, (2.39)
wherez(t)is the flow of the autonomous vector field, Z =X+1
22 Z t
0
Xτdτ , Xt
, (2.40)
with the initial conditionz(0) =x0. The beauty of the series expansions approach lies in the fact that no new theorems need to be derived to obtain and analyze the consquences of the second-order average.
Equation (2.39) is close to what Sanders and Verhulst obtain for the compensating flow. Sanders and Verhulst also require the integral term in the flow to have a vanishing average [161]. Bogoliubov and Mitropolsky [17] obtain the same zero average assertion. By introducing an integration constant to ensure a zero average for the termRt
0 X(y(t), τ)−X(y(t))
dτ, the integration constant factors out the effect of varying the initial timet0 for which the monodromy map in computed, c.f. Section 2.3.2. For systems that are asymptotically stable, the transient dynamics related to the initial conditions of the full system (2.21) are neglected. Examples in Sections 3.3 and 5.3 address this issue, and also relate it to Theorem 10.
Remark. There are significant differences between the generalized averaging theory and classical averag- ing theory in spite of their similarity. In order to utilize the powerful results of the chronological calculus, the vector fields must be smooth or analytic. The pertubation approach of Sanders and Verhulst requires the existence of a first derivative and Lipschitz continuity up to the first derivative affording it a level of gener- ality that does not hold for the former. The same goes for the improvednth-order averages found in Ellison et al. [42]. Truncated expansions may be calculated without regard to the requirements of the chronological calculus. By appealing to the classical averaging theorems it is possible to determine the minimal require- ments for the calculated average to hold. Regardless of the requirements, it is still posible to carry out the formal series expansions, whose truncations may provide asymptotic understanding of the actual flow [145].
2.4.1 Higher-order Averaging
Third-order averaging. The truncated Floquet mapping for third-order averaging is Trunc2(P(t)) =Id +
Z t 0
Xτ −X
dτ + 1 22
Z t 0
Z τ 0
Xsds , Xτ
− Z t
0
Xsds , Xt
! dτ +1
22 Z t
0
Xτdτ ◦ Z t
0
Xτdτ −2 Z t
0
Xτdτ ◦X t+1
22X◦X t2
(2.41) This truncation satisfiesP(0) =P(T), but may not periodic fort > T. If this is so, then it may be amended according to the discussion in Section 2.3.4. The autonomous averaged vector field is
Z =X+1 22
Z t 0
Xτdτ , Xt
+1 4T 3
"
X, Z t
0
Xτdτ , Xt
# +1
33 Z τ
0
Xτ1dτ1, Z τ
0
Xτ1dτ1, Xτ
(2.42) Fourth-order averaging. The fourth-order averaging results begin to get complicated due to the rate of growth of the averaging terms with each higher order. Averaging calculations for higher orders increasingly become computationally expensive. This fourth level is the last averaging order that will be computed here.
In the sequel is given the general algorithm followed in case fifth- or higher-order averaging is required.
The Floquet mapping for fourth-order averaging can be found in Table 2.1, and the autonomous averaged vector field is in Table 2.2. The truncation is easily amended to beT-periodic.
2.4.2 A General Averaging Algorithm
Although calculating higher-order expansions for averaging is a difficult task, there is a very simple algo- rithm for doing so. Based on the results from the previous sections, this algorithm will obtain a method
Trunc3(P(t)) =Id + Z t
0
Xτ −X dτ +1
22 Z t
0
Z τ 0
Xsds , Xτ
− Z σ
0
Xsds , Xσ
! dτ +1
43 Z t
0
Xτdτ , Z t
0
Z τ
0
Xsds , Xτ
dτ
−T
"
X, Z t
0
Xτdτ , Xt
# t
!
+1 33
Z t
0
Z τ 0
Xsds , Z τ
0
Xsds , Xτ
− Z t
0
Xτdτ , Z t
0
Xτdτ , Xt
! dτ +1
22 Z t
0
Xτdτ ◦ Z t
0
Xτdτ −2 Z t
0
Xτdτ ◦X t+ 1
22X ◦X t2 +1
43 Z t
0
Xτdτ ◦ Z t
0
Z τ 0
Xsds , Xτ
dτ − 1 22
Z t 0
Xτdτ ◦ Z t
0
Xτdτ , Xt
t
−1 23
Z t
0
Z τ 0
Xsds , Xτ
dτ ◦X t+ 1 43X ◦
Z t 0
Xτdτ , Xt
t2
+1 43
Z t 0
Z τ 0
Xsds , Xτ
dτ ◦ Z t
0
Xτdτ + 1 43
Z t 0
Xτdτ , Xt
◦X t2 +1
63 Z t
0
Xτdτ ◦ Z t
0
Xτdτ ◦ Z t
0
Xτdτ ,−X◦X ◦X t3
Table 2.1: Truncated Floquet mapping,Trunc3(P(t)), for fourth-order average.
of averaging that contains two dimensions of approximation. The first is in the truncations of the loga- rithm vector field,Truncm(Z), which captures up tomth-order the dynamics of the system. The second is the truncations of the compensation map,Trunc(m−1)(P(t)), which will give mth-order proximity of the composed system to the actual system.
The two components of the generalized averaging theory lead to a very important distinction that can sometimes be confused when performing higher-order averages. For example, ignoring the Floqut mapping and using only the autonomous flow results in averaged equations of motion that are capable of capturing themth-order dynamics, but will only be good to1st-order proximity. The distinction between roles of the mappingP(t)and the autonomous vector field was also made by Bogoliubov and Mitropolsky [17].
The averaging process described by Bogoliubov and Mitropolsky utilizes themth-order average to find the(m + 1)th-order average. This is done by viewing the solution to the(m + 1)th-order average as coming from a perturbed version of themthorder average. The technique is akin to a Picard iteration on the averages.
The series expansions given in this dissertation and elsewhere are an attempt to extract structure from the Picard iterations (a.k.a Volterra series expansions). An important similarity is that the higher-order averages build upon the lower-order averages in a systematic way.
The averaging method provided here is capable of replicating classical averaging results and appears to coincide with known methods for obtaining higher-order averages. The general strategy implied by the averaging theory derived herein can be found in Table 2.3.
Z =X+1 22
Z t 0
Xτdτ , Xt
+1 4T 3
"
X, Z t
0
Xτdτ , Xt
#
+1 33
Z τ 0
Xτ1dτ1, Z τ
0
Xτ1dτ1, Xτ
+ 1 33
Z τ 0
Xτ1dτ1, Z τ
0
Xτ1dτ1, Xτ
+1
33[a1, a21] + 1
33
"
a1, Z τ
0
Xτ1dτ1, Xτ
# + 1
33
"Z τ 0
Xτ1dτ1, a21
#
− 1 124
Z τ 0
Z τ1
0
Z τ2
0
Xτ3dτ3, Xτ2
dτ2,[Xτ1, Xτ]
dτ1
− 1 124
Z τ 0
Z τ1
0
Z τ2
0
Xτ3dτ3, Xτ2
dτ2, Xτ1
dτ1, Xτ
− 1 124
Z τ 0
Z τ1
0
Xτ2dτ2, Z τ1
0
Xτ2dτ2, Xτ1
, Xτ
dτ1
Table 2.2: Truncated averaged vector field,Trunc4(Z), for fourth-order average.
1. Calculate the logarithm vector field,Truncm(Z).
2. Calculate the truncated version ofexp (−Zt).
3. Calculate the truncated version of−→expRt
0Xτdτ . 4. Use the truncations forTrunc(m−1)(P(t)).
Table 2.3: Algorithm for Computing the Average.