3.1. Motivation
In our previous work [
9], it was assumed that the invariant submanifold,
, can be described by the following series:
where
are the undetermined coefficients. Then, the problem reduces to finding the correct value of these coefficients, which can be found using the quadrature method. The proposed solution (
6) can be substituted in Equation (
4) to simplify the calculations of the correct values of coefficients
. This solution can be rewritten in matrix form as
where
and
Two issues must be considered here. First, in certain cases series (
6) might diverge, depending on the frequency and amplitude selected. This situation usually happens in LAOs. Notice that series (
6) is equivalent to
where
; therefore, although the series coefficients are convergent, i.e.,
, the overall series coefficients for high order powers, i.e.,
for
and
, might not converge for high amplitudes, i.e.,
. Second, where the series does not diverge, the higher order coefficients have the same order of magnitude as those with lowerr order power, i.e.,
. Hence, the truncated series is useless for computation.
For instance, in [
9], the following dynamical system was analyzed:
where
is the state variable and
is the input variable. When system (
10) is subjected to an oscillatory input
, where
is the first element of the state vector of the oscillator (
2)–(
3) past a transient behavior that depends on the initial conditions, the system will approach an invariant submanifold
that satisfies the following nonlinear partial differential equation:
Then, applying the method proposed in [
9], the following series solution with the form in (
6) can be obtained:
The terms of order
of this series for
, are proportional to
; as such, in [
9] it was shown that an approximated limit to guarantee the convergence of series (
12) is that the frequency–amplitude relation must satisfy
. When this limit is not fulfilled, the solution (
12) is no longer suitable to describe the dynamical behavior and the series may diverge, as the series coefficients for high order powers are not convergent and may have the same order of magnitude as the terms with low order power. For instance, when the system is in LAO mode, by comparing the first and the last term of series (
12),
and
, respectively, it is clear that, for
, the 7-th order term of
has a larger amplitude than the first order term of
. In this context, the system is in LAO mode. For this reason, the present work proposes a modification of series (
6) to avoid this inconvenience.
3.2. Proposed Series Solution
Due to the trigonometric identity
certain terms with high order power can be represented as a combination of smaller order powers. For instance, vector
is equivalent to
Therefore,
can be reformulated as a function of
and
, eliminating the factors of
with order powers larger than 1. For instance, the term
is equal to
, and the original coefficient,
, for
, now has an extra contribution in the form
that can have the same order of magnitude as
and depends on the amplitude. For instance, using identity (
14), the series solution (
12) is equivalent to
except now the coefficients of the series are functions of both the frequency and amplitude, and each of these coefficients can be considered as an infinite series.
Following this approach, by replacing vector (
14) in Equation (
7) it is straightforward to verify that the series (
6) is equivalent to
where
and
are undetermined coefficients that depend on both the frequency and amplitude, in contrast with the coefficients
of series (
6), which only depend on the frequency. In matrix form, the vector
becomes
where
The new coefficients are now a function of both the frequency and amplitude, and can be considered as an infinite series of the previous coefficients .
Note that instead of replacing
in Equation (
14) to obtain the series (
16) with terms with higher than first-order powers only for
, a similar analysis can be carried out by replacing
in vector
to obtain an equivalent series with higher order than the first order terms, now in
. This case is briefly described in
Appendix A.
The use of the matrix form (
17) allows us to simplify both the analysis and solution. The following two operational definitions are instrumental in finding the series (
16) that satisfies Equation (
4).
Definition 1. Given vectors and with the same dimension (where N can be infinity), the product operator is defined aswhere the elements of matrix P are Definition 2. Given vectors and with dimension , the differential operator associated with the oscillation with frequency ω and amplitude A is defined aswhereand the elements of matrix D arewhile the elements of matrix P are defined in Equation (20). To substitute the proposed solution (
16) in Equation (
4), it is necessary to compute the time derivative of
; considering that
is the sum of coefficients of the form
and
, with
, their time derivatives are
Furthermore, considering the dynamics of
z, Equations (
2) and (
3) and the trigonometric identity provided in Equation (
13), it holds that
Following this structure, the time derivative of the infinite series (
16) or its matrix form, (
17), is presented in the following Proposition.
Proposition 1. Consider vectors and of infinite dimension with constant coefficients and associated with a function of and , defined aswhere vector is defined in Equation (8). The time derivative of iswhere is the operator described in Definition 2. Proof. The time derivative of
is
Notice that the derivative of vector
with respect to
is
where
is defined in Equation (
22). Therefore, the partial derivatives of
with respect to
and
are
therefore, the time derivative of
becomes
due to the trigonometric identity (
13) and the fact that
where
is defined in Equation (
20); the time derivative of
simplifies to
Finally, with Definition 2, it holds that , concluding the proof. □
The time derivative presented in Proposition 1 includes the differential operator
, defined in Equation (
21), and the matrix
with a tridiagonal structure, the elements of which are
producing a shift in the coefficients of the series. Then, expressing the time derivative of
as
, the coefficients for
of
,
and
contain the coefficients
and
of
.
Furthermore, as Equation (
1) contains nonlinear terms, it might have products of finite or infinite series of
and
, such as
, where
can be expressed in matrix form as
where
while the procedure to compute the product of two series is presented in the following proposition.
Proposition 2. Consider vectors , , , and with dimension () associated with functions and of and , defined asthe product of these functions iswhich is equivalent towherewhile the operatoris described in Definition 1, P is defined in Equation (20), andis the identity matrix of dimension N.
Proof. As
is a scalar function, Equation (
40) is equivalent to
For any pair of vectors
and
the product
is
and given the property (
34), this product is equal to
where
is provided in Definition 1. Using this formula and the trigonometric identity (
13), the product (
44) becomes
Finally, property (
34) is applied again to obtain
which is equivalent to Equation (
41), concluding the proof. □
It is important to note that the time derivative and the product described in Propositions 1 and 2 have the same structure as series (
17). Hence, it is possible to establish the conditions to find the oscillatory invariant submanifold (
4) presented in the following theorem.
Theorem 1. Considering the input-state stable differential Equation (1) with and along with the dynamics defined in Equations (2) and (3), assume that reaches an invariant submanifold that satisfies the equations Then, if functions can be expressed as series of integer powers of the elements of x and z, the solution of each element of vector ξ can be proposed to be an infinite series of integer powers of and , i.e.,which in matrix form iswhere vectorsare the rows of matrix , defined in Equation (18). Proof. The system of differential equations defined in Equation (
45) is
Assume that the elements of the invariant submanifold
can be described as in Equation (
47); then, according to Proposition 1, their time derivatives are
where
is provided in Definition 2. Using this time derivative, the elements of Equation (
45) become
for
. If functions
can be expressed as series of integer powers of the elements of
x and
z, then using the identities in Equations (
36) and (
41) in Proposition 2, the differential equations can be transformed to the form
where
and
are vectors that depend on the values of coefficient matrices
and
, defined in Equation (
18). Independently of the values of vectors
and
, Equation (
49) holds if
for
. Then, the problem reduces to finding the coefficients
,
,
,
by solving Equations (
50) and (
51) to obtain the solution in Equation (
47). □
Equations (
50) and (
51) imply that there are two set of equations for coefficients with the same order, namely, those associated with
(Equation (
50)) and those associated with
(Equation (
51)); therefore, in order to reduce the degrees of freedom it is necessary to consider the initial conditions
, where
is such that
and
. Then, at
it holds that
while the value of
at
is
Therefore, the first
n parameters of both
and
in series (
7) are correlated, and their value depends on the initial conditions
To illustrate the proposed method, in the following section the same study cases used in our previous contribution [
9] are analyzed.