Boundary-layer Features in Limit of Conservative Nonlinear Oscillators

In the double limit of high amplitude (xmax  ∞) and high leading power (x , N  ∞), (1+1) dimensional conservative nonlinear oscillatory systems exhibit characteristics akin to boundary layer phenomena. The oscillating entity, x(t), tends to a periodic saw-tooth shape of linear segments, the velocity, x(t), tends to a periodic step-function and the x  x phase-space plot tends to a rectangle. This is demonstrated by transforming x and t into proportionately scaled variables,  and , respectively. () is (2-) periodic in  and bounded (|()| ≤ 1). The boundary-layer characteristics show up by the fact that the deviations of (), () and the    phase-space plot from the sharp asymptotic shapes occurs over a range in  of O(1/N) near the turning points of the oscillations. Email: zarmi@bgu.ac.il Orcid: 0000-0001-8830-9667


Introduction
There are hundreds of papers, dozens of textbooks, lecture notes and research theses that discuss the periodic behavior of linear systems, to which a small nonlinear perturbation is added, through a perturbative analysis. However, there is no satisfactory methodology for the study of such systems in the large-amplitude limit, when the nonlinear perturbation is large compared to the linear part of the equation. Absence of such methodology may be a consequence of the fact that high amplitudes go hand in hand with high acceleration. For an oscillating mass, the classical description then ceases to be valid; relativistic effects enter.
However, there are many applications in Physics (e.g., current oscillations in electronic devices), Chemistry and Biology (e.g., oscillations in chemical reactions) and Engineering (e.g., oscillations of mechanical systems), where non-relativistic large-amplitude oscillations are of relevance. The literature has focused mainly on large-amplitude oscillations of periodically driven oscillatory systems, in connection with resonance phenomena or chaotic behavior. This paper focuses on the large-amplitude and high exponent limits of solutions of conservative systems of linear oscillators, to which nonlinear terms are added. Closed-form solutions have been derived the cases of the Duffing and Quintic oscillators [1][2][3][4][5][6]. There are works that have dealt with such systems through various expansion methods . In all of the approximation approaches, the validity of the numerical approximations to the solution is limited in time, typically, to several periods, because the period is computed with some approximation method. To see how this problem arises, consider a nonlinearly perturbed harmonic oscillator: , ( where f(x) is a nonlinear function of x(t). A perturbation analysis can be developed as long as . ( When the amplitude of oscillations, xmax, is very large, the magnitude of f(x) can be much greater than that of the linear term. Consider the following equation of motion: .
Often, for leading nonlinear term, (a2 N+1 x 2 N+1 ), one assumes a2 N+1 =  « 1. However, when the maximal amplitude, xmax is very large, the magnitude of  is irrelevant. What counts is the fact that the leading term is much greater than the next-to-leading term, (a2 N1 x 2 N1 ). For large xmax, the maximal velocity of oscillations, vmax, and the period of oscillations, T, vary as: As a result, the characteristics of the solution for large xmax are determined by a simpler equation, which provides an approximation that improves as xmax grows: .
The solutions of the full equation, Eq.
(3) and of Eq. (5) will be analyzed around the (largeamplitude limit) solution by an expansion in terms of a small parameter, : .
Using Eq. (6), the general expression for the period, T, has the form: .
In the large-N limit, one has . We shall be interested primarily in G0, the large-amplitude limit of G. As the discussion is limited to conservative systems, T can be computed. In the cases of N =1 and 3 (cubic and quintic oscillators, respectively), the large-amplitude limit of T has the form: .
With more complicated nonlinearities, T can be computed numerically to any desired accuracy. G0, the leading term in G is O( 0 ). The next-to-leading term in the expression in G is determined by the next-to-leading term in Eq. (3). If that term is a2 q+1 x(t) 2 q-1 (q < N), the deviation of G from G0 is O( 2 (N-q) ). In the case of the simpler Eq. (5), the deviation of G from G0 is O( 2 N ). In the case of the cubic oscillator (Duffing), the correction is O( 2 ).
In standard expansion approaches, the period is approximated through a perturbation scheme.
From Eq. (4), it is clear that x(t) and x(t) vary rapidly between zero and exceedingly large values over a very short time span. A perturbation-type approximation is bound to produce results of limited validity in time. A better way to study the large-amplitude and large-N limits is based on rescaling the x(t) and t: .
. This is studied in Section 4.

The Duffing equation
The solution of the Duffing equation, .
is known in closed form [1][2][3][4][5][6]. Its analysis in the formulation offered here is performed as a simple example for the procedure.
The total energy is given by: .
Eq. (15) yields the expression for the period of oscillations: , where  is defined in Eq. (6) and K(k) is the complete elliptic integral of the second type [36], .
Note that the relative deviation of T from the large-amplitude limit (  0) is O( 2 ), which is nothing but the order of magnitude of the maximal ratio of the linear term to the cubic term in Eq. (13). As xmax becomes large, one has .
Thus, the maximal amplitude and velocity grow while the period decreases.
the spectrum is dominated by the first two Fourier components: a1 = 0.9550 and a3 = 0.0430.
The next Fourier coefficient a5 is already 0.0019. .

The limits of xmax and
Exploiting Eq. (16), the large-N limit, Eq. (23) is found to be: .
A detailed numerical analysis reveals that the constant in large-N expression for matching is close to 2.
Thus, as the power, (2 N + 1), grows, the range in , over which 0() deviates from the linear behavior diminishes. In summary, 0() is almost linear in . It bends nonlinearly to ± 1 close to the turning points. As a result, the profile of 0() is almost constant at ±vmax, and changes sign over a short range in  around the turning points. Namely, as N grows, 0() approaches a periodic step function. This is demonstrated in Fig. 2. 0() approaches a peri-odic saw-tooth shape with linear segments, demonstrated in Fig. 3. Finally, thanks to the  dependence of 0() and 0(), the phase-space plot of 0() vs. 0() approaches a rectangle as N is increased, demonstrated in Fig. 4.

General N
The zero-order approximation to (), the solution of Eq. (25), when the latter is expanded in a power series in , is 0(), the solution of Eq. (27). It is the infinite-amplitude limit of (). from the asymptotic solution, 0(), becomes progressively smaller as N is increased.

The Duffing equation: Perturbation expansion for large but finite xmax
The Duffing equation offers an exception. In the large-amplitude limit, both the term in Eq. . This is demonstrated in Fig. 5. In the case of (), there is a small discrepancy between the full solution and the zero-order approximation. The situation is remedied by the effect of the second-order contribution of 2(), as demonstrated in Fig. 6. In summary, the quality of the second-order approximation is excellent although the cubic term is greater than the linear term in the equation by no more than a factor of .

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The author declares that no funds, grants, or other support were received during the preparation of this manuscript.
The author has no relevant financial or non-financial interests to disclose.
1) The author made all contributions to the conception or design of the work. No data acquisition was required. No new software was created.
2) The author drafted the work or revised it critically for important intellectual content; 3) The author approved the version to be published; and 4) The author agrees to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved.

Data availability Statement
Data sharing is not applicable to this article, as no datasets were generated or analyzed during the current study.        Figure 1 Large-amplitude limit G 0 (Eq. (7)) (red); Large-amplitude and large-N limit of G 0 (Eq. (8)) (blue).