This paper focuses on the characteristics of solutions of nonlinear oscillatory systems in the limit of very high oscillation energy, E; specifically, systems, in which the nonlinear driving force grows with energy much faster for x ( t ) close to the turning point, a(E), than at any position, x(t), that is not too close to a(E). This behavior dominates important aspects of the solutions. It will be called “nonlinear violence”. In the vicinity of a turning point, the solution of a nonlinear oscillatory systems that is affected by nonlinear violence exhibits the characteristics of boundary-layer behavior (independently of whether the equation of motion of the system can or cannot be cast in the traditional form of a boundary-layer problem.): close to a(E), x(t) varies very rapidly over a short time interval (which vanishes for E → ∞). In traditional boundary layer systems this would be called the “inner” solution. Outside this interval, x(t) soon evolves into a moderate profile (e.g. linear in time, or constant)—the “outer” solution. In (1 + 1)-dimensional nonlinear energy-conserving oscillators, if the solution is reflection-invariant, nonlinear violence determines the characteristics of the whole solution. For large families of nonlinear oscillatory systems, as E → ∞, the solutions for x(t) tend to common, indistinguishable profiles, such as periodic saw-tooth profiles or step-functions. If such profiles are observed experimentally in high-energy oscillations, it may be difficult to decipher the dynamical equations that govern the motion. The solution of motion in a central field with a non-zero angular momentum exhibits extremely fast rotation around a turning point that is affected by nonlinear violence. This provides an example for the possibility of interesting phenomena in (1 + 2)-dimensional oscillatory systems.
There are hundreds of papers, books, theses and lecture notes that deal with the analysis of linear oscillatory systems, which are affected by a small nonlinear perturbation. The pioneering works of Poincaré [
x ¨ ( t ) + c x ( t ) | x ( t ) | k , c > 0 , k > 0 (1)
have been developed in terms of the inverse of the incomplete Beta function (Ateb(h) functions) [
In the procedures exploited in these references the amplitude, x(t), and the period, T, are approximated by a power expansion in a small parameter through a finite order. The approximation for T leads to the emergence of secular terms, which spoil the quality of the approximation to the solution if computed for many cycles of oscillation. However, secular terms can be avoided in the analysis of energy-conserving oscillatory systems in (1 + 1) and some (1 + 2)-dimensional systems, as the period of oscillations, T, is then either known in closed form (e.g., central-field motion in a plane [
This study focuses on the behavior of nonlinear oscillatory systems in the limit of extremely high oscillation energy, E (mathematically, E → ∞), when the nonlinear driving force grows with E much faster for x(t) close to the turning point, a(E), than at any position, x(t), that is not too close to a(E). In many such systems, this behavior of the driving force determines important aspects of the solution in the limit E → ∞. In some systems it even determines the full profile of the solution in that limit. Hence, such dominance of the nonlinear part of the driving force will be called “nonlinear violence”. The main message is that when nonlinear violence dominates, the solution develops boundary-layer characteristics, which emerge independently of whether the equation of motion of the oscillator can or cannot be transformed into the traditional form of a boundary-layer system—a boundary-value problem described by an ODE, in which the highest derivative is multiplied by a small parameter, e.g.:
ε x ¨ ( t ) + x ( t ) + g ( x ( t ) , x ˙ ( t ) ) = 0 , | ε | ≪ 1 , x ( t 1 ) = x 1 , x ( t 2 ) = x 2 (2)
A boundary-layer analysis has been applied in hundreds of papers to the study of oscillation-relaxation systems. For the study of such problems, see, e.g., [
One purpose of this paper is to show that solutions of nonlinear oscillatory systems that are dominated by nonlinear violence exhibit characteristics that are similar to those of boundary-layer problems. When nonlinear violence occurs in the vicinity of a turning point, the solution varies rapidly around that point over a short time interval, the length of which tends to zero as E → ∞. In boundary-layer terminology, this would be called the “inner solution”. Beyond this interval, the solution rapidly evolves into a moderate profile, often, well approximated by linear time dependence, in other systems—by a constant. In boundary-layer terminology, this would be called the “outer solution”. Exploiting the analysis tools developed in boundary-layer theory facilitates understanding of the characteristics of the solutions, in particular, in the limit of very high oscillation energy.
Section 2 deals with (1 + 1)-dimensional conservative systems in the limit of total energy, E → ∞. Systems not transformable to boundary-layer formulation are analyzed in Section 2.1. The first to be discussed, are systems, the solutions of which are symmetric under space inversion and tend to saw-function patterns. The general case of potential energy, V(x), that is symmetric under reflection in x, when V(x) grows sufficiently rapidly near the turning points, x = ±a(E) is studied in Section 2.1.1. An example of a Gaussian potential is presented in Section 2.1.2. The case of a polynomial driving force with constant coefficients with a positive coefficient multiplying the leading power is reviewed in Section 2.1.3. The case of the extreme relativistic limit of the relativistic harmonic oscillator is discussed in Section 2.2. Unlike all previously discussed systems, in the extreme relativistic limit, the dynamical equation of the relativistic harmonic oscillator is transformable to a boundary-layer problem. In all the system discussed, the solutions are governed by nonlinear violence. The growth characteristics of V(a(E)) lead to a profile of x(t), which, to a good approximation, is comprised of rising and decreasing segments with linear time dependence throughout most of each half of the period, so that x ˙ ( t ) is very close to a constant value rapidly changing sign over the vanishingly small interval around the turning points. Around the turning points, the solution is highly curved. Consequently, As E → ∞, x(t) tends to a periodic saw-tooth profile, the velocity, x ˙ ( t ) ,—to a periodic step-function, the acceleration, x ¨ ( t ) ,—to a periodic sequence of δ-functions and the phase-space plot, x ˙ ( t ) vs. x(t),—to a rectangle.
A different effect of nonlinear violence emerges in some systems with polynomial nonlinearities, in which the coefficients of one or more of the lower powers in the polynomial grow with the maximal amplitude, a(E), so rapidly that they compete with the effect of the leading power of the polynomial when the solution, x(t), is close to a(E). For such systems, x(t) tends to a step function when a(E) grows indefinitely. The Duffing equation, in which the coefficient of the linear term grows with energy, is discussed in Section 2.3 because of its special physical significance. It was obtained by an ingenious transformation of the relativistic harmonic oscillator [
The case, in which V(x) is not symmetric under reflection in x, and nonlinear violence emerges only near one turning point is discussed in Section 3.
The motion of a particle in a central field, V(r) (r is the distance of the particle from the origin) in (1 + 2) dimensions, when E → ∞ and the particle has non-zero angular momentum, opens the door to interesting phenomena in higher space dimensions. It is discussed in Section 4. The classical case of motion under a gravitational field, V(r) µ −1/r, follows a single “oscillation” with an infinite period of motion [
In most systems discussed, when a turning point that is affected by nonlinear violence is crossed (crossing time t0), as E → ∞, the velocity jumps by a finite value, Δv. As a result, the acceleration tends to Δv ∙ δ(t − t0). For each system discussed, it is shown that, within the accuracy of calculational procedure, this jump is reproduced by the “inner solution” of the boundary layer-like analysis.
Consider an energy-conserving oscillatory system:
x ¨ ( t ) + V ′ ( x ( t ) ) = 0 (3)
where V(x) → ∞ as x → ∞ and the maximal amplitude, a(E) tends to infinity as the total energy, E → ∞. To ensure oscillatory behavior, V(x), must be positive at both turning points. Nonlinear violence emerges in the vicinity of a turning point, x = a(E), if V(x) grows very rapidly in the vicinity of that point.
To see how a boundary layer-like behavior emerges when V(x) is symmetric under reflection in x, scaled coordinates will be used:
x ( t ) = a η ( θ ) , θ = 2 π t T (4)
The motivation for the choice of the angle variable, θ, is that the motion is periodic with period T. As T can be found either in closed form, or computed to numerically to any desired accuracy (see discussion in Section 1), in the analysis presented here, T is treated as a known constant. Furthermore, as the oscillations vary in the range −a ≤ +a, to obtain a standard limiting profile form that applies to many systems, the coordinate, x(t) is replaced by the bounded scaled coordinate, η, −1 ≤ η ≤ +1, with period (2π) in θ. Equation (3) becomes a boundary value problem for η:
η ″ ( θ ) + 1 a ( T 2 π ) 2 V ′ ( a η ( θ ) ) = 0 , η ( 0 ) = − η ( π ) = η ( 2 π ) = 1 (5)
The expression for the period, T is:
T = 4 ∫ 0 a ( 1 / 2 ( E − V ( x ) ) ) d x = a ( 8 / V ( a ) ) I I = ∫ 0 1 ( 1 / 1 − ( V ( a η ) / V ( a ) ) ) d η (6)
Often the integral, I, defined in Equation (6) is O(1). For example, if the potential function obeys the requirement:
V ( a η ) / V ( a ) → a → ∞ | η | < 1 , fixed C < 1 (7)
where C is a constant, independent of a, then
I → a → ∞ 1 1 − C (8)
In many physical systems, C = 0, and then I → 1. When Equation (7) holds, T vanishes in proportion to (V(a))−1/2 = E−1/2.
Exploiting Equation (6), Equation (5) becomes:
η ″ ( θ ) + 2 I 2 π 2 a V ′ ( a η ( θ ) ) V ( a ) = 0 (9)
When nonlinear violence dominates, Equation (9) leads to the emergence of behavior of the solution, which looks like the “inner” and “outer” solutions of boundary-layer problems. This happens if
I 2 a V ′ ( a η ) V ( a ) → a → ∞ | η | < 1 , fixed 0 (10)
I 2 a V ′ ( a ) V ( a ) → a → ∞ ∞ , (11)
Note the dramatic difference between Equations (10) and (11). Equation (10) ensures that, as long as |η| < 1 (specifically, (1 − |η|) >> O(1/a)), namely, not too close to the turning points, the second term in Equation (9) becomes negligible, so that η ″ ( θ ) is very small. Thus, Equation (11) ensures that, except near the turning points, Equation (9) tends to
η ″ ( θ ) = 0 , ( a → ∞ , | 1 − η | ≫ O ( 1 / a ) ) . (12)
Equation (11) yields a linear profile for the “outer solution”:
η outer ( θ ) = A θ + B . (13)
Consider the interval, 0 ≤ θ ≤ π, with initial conditions: η(0) = 1 and η'(0) = 0. Owing to symmetry and periodicity of the solution, one has ηouter(π/2) = 0. Also, although ηouter may miss the actual solution very close θ = 0 and π, up to errors of at most O(1/a), it must reach close to +1 as θ → 0 and -1 as θ → π: Hence, Equation (13) must be:
η outer ( θ ) ≈ A ( π / 2 + δ ( a ) − θ ) , A = ( 2 / π ) + ε ( a ) , δ ( a ) , ε ( a ) → a → ∞ 0 (14)
On the other hand, Equation (11) ensures that, very close to the turning point, η ″ ( θ ) is very large, so that the maximum η(0) = +1 is very sharp. η(θ) varies rapidly over a very short range in θ. Hence, θ = 0 is the tip of a boundary layer. As η(0) = +1 is a point of a symmetric maximum for η(θ), approximate η(θ) for small θ by
η inner ( θ ) = 1 − b θ 2 + c θ 4 + O ( θ 6 ) . (15)
Choosing the nomenclature “inner” for this part of the solution is based on the observation presented below, that it varies rapidly over a narrow range in θ in the vicinity of θ = 0. This is just the characteristic of the inner solution in a boundary-layer problem. Replace η(θ) in Equation (9) by ηinner(θ) and expand the result around θ = 0 through θ4 to find:
b = I 2 π 2 a V ′ ( a ) V ( a ) , c = O ( ( I 2 a V ′ ( a ) V ( a ) ) 2 ) . (16)
Requiring that the inner and outer solutions coincide and so do their derivatives at a matching point, θm, one obtains for θm, and for A (see Equation (13)):
θ m = π I 2 V ( a ) a V ′ ( a ) + O ( ( V ( a ) a I 2 V ′ ( a ) ) 2 ) A = 2 π ( 1 + 1 I 2 V ( a ) a V ′ ( a ) + O ( ( 1 I 2 V ( a ) a V ′ ( a ) ) 2 ) ) . (17)
Equation (11) ensures that the range in θ, over which the “inner solution” evolves, is small and vanishes for a → ∞. Consequently, the matching point, θm of Equation (17), shrinks to zero and, concurrently, that A tends to (2/π) as a → ∞. With Equations (10) and (11) satisfied, as a → ∞, η(θ) tends to a periodic saw-tooth profile: Linear segments connect the turning points (η(2nπ) = +1, η((2n + 1)π) = −1), the velocity, η'(θ), tends to a periodic step-function profile, alternating between ±(2/π) and the phase-space plot, η'(θ) vs. η(θ), tends to a rectangle.
δ-function profile for acceleration The periodic step-function limit of the profile of η'(θ) as a → ∞ implies that, η''(θ), tends to a periodic sequence of δ-functions:
η ″ ( θ ) → a → ∞ ( − 1 ) n + 1 4 π δ ( θ − n π ) . (18)
The δ-functions arise from the cumulative change in velocity, η', across a turning point. Within a very a good approximation, the contribution of the “inner solution” generates the jump in η' as the turning point is crossed, as a → ∞. Using ηinner(θ) of Equation (14) through O(θ2) for η(θ) as well as Equations (15) and (16), yields the total change in the slope:
η ′ ( θ m ) − η ′ ( − θ m ) = ∫ − θ m + θ m η ″ ( θ ) d θ ≈ ∫ − θ m + θ m η ″ inner ( θ ) d θ = − 2 b θ m = − 4 π . (19)
Note that the result of Equation (19) does not depend on the asymptotic behavior of the period, T or the integral I (see Equation (6) and the following discussion).
As an example to the statements made in Section 2.1.1, consider a Gaussian potential,
V ( x ) = 1 2 e x 2 (20)
Equations (10) and (11) are satisfied for this potential as a → ∞.
Using the scaled coordinates of Equation (4), the period is given by:
T = 4 a e − a 2 / 2 I , I = ∫ 0 1 ( 1 / 1 − e − a 2 ( 1 − η 2 ) ) d η = 1 + e − a 2 E r f [ i a ] 4 i a + ⋯ → a → ∞ 1 (21)
For the error function, see [
The equation for η(θ) is:
η ″ ( θ ) + 4 a 2 I 2 π 2 e − a 2 ( 1 − η ( θ ) 2 ) η ( θ ) = 0 . (22)
Figures 1-3 present numerical solutions of Equation (22), showing, respectively, how η(θ) tends to a periodic saw-tooth profile, η'(θ)—to a periodic step-function and the phase plot, η'(θ) vs. η(θ), to a rectangle. Let us see how this happens.
At the turning point |η| = 1 and |η'| = 0, energy conservation for Equation (22) yields:
η ′ ( θ ) = ± 2 π 1 − e − a 2 ( 1 − η ( θ ) 2 ) I . (23)
Equation (23) implies that max|η'(θ)| (attained at θ = π/2 and 3π/2, where η(θ) = 0), is:
max | η ′ ( θ ) | = 2 π 1 − e − a 2 I → a → ∞ 2 π . (24)
The limit of Equation (24) also holds for η'(θ) of Equation (23) as long as (1 − η(θ)2) = O(1):
η ′ ( θ ) → a → ∞ 1 − η ( θ ) 2 = O ( 1 ) = ± 2 π . (25)
Thus, except near the turning points, where η(θ) ≈ ±1, the profile of the “outer solution” tends to a straight line, which, owing to the symmetry and periodicity of the solution, must pass through zero at θ = π/2:
η outer ( θ ) → a → ∞ 1 − η ( θ ) 2 = O ( 1 ) ± 2 / π ( π / 2 − θ ) . (26)
Only near the turning points, for (1 − η(θ)2) = O(1/a2), is this limit not attained, and the “inner solution” evolves. For η(0) = 1 and η'(0) = 0 at the turning point at θ = 0, the solution is maximal and symmetric in deviations around θ = 0. Hence, expand η(θ):
η inner ( θ ) = 1 − b θ 2 + O ( θ 4 ) . (27)
Substituting Equation (27) in Equation (22), and expanding the equation through the O(θ2) yields:
b = e a 2 T 2 8 π 2 = 2 π 2 a 2 I 2 . (28)
η ′ inner ( θ ) = − 4 a 2 I 2 π 2 θ + O ( θ 3 ) . (29)
Using Equations (21) and (27)-(29) yields the “inner solution”, whereas the “outer solution” is determined up to an overall coefficient:
η inner = 1 − 2 a 2 I 2 π 2 θ 2 + O ( θ 4 ) , η outer = A ( π 2 − θ ) . (30)
Requiring that the two solutions and their derivatives coincide at a matching angle, θm, in the vicinity of θ = 0, one obtains:
θ m ≈ a ≫ 1 π 2 a 2 I 2 ( 1 + O ( 1 / a 2 ) ) , (31)
A = 2 π ( 1 + O ( 1 / a 2 ) ) . (32)
θm, the range, over which the “inner solution” evolves, shrinks to zero and A tends to (2/π) at an O(1/a2) rate.
In summary, as a → ∞, the solution for η(θ) tends to a periodic saw-tooth profile, which varies linearly from +1 at θ = 0, to −1 at θ = π and back to +1 at θ = 2π. η'(θ) tends to a periodic step function, with values of −(2/π) over 0 ≤ θ ≤ π and +(2/π) over π ≤ θ ≤ 2π.
δ-function profile for acceleration As in Section 2.1.1, η''(θ) tends to a periodic sequence of δ-functions with alternating signs (Equation (18)). Again, the jump in η'(θ) as the turning point is crossed is well approximated by the cumulative change of η'(θ) over the “inner solution”. Substituting ηinner(θ) of (24)-(29) for η(θ) through O(θ2) one obtains:
η ′ ( θ m ) − η ′ ( − θ m ) ≈ ∫ − θ m + θ m η ″ inner ( θ ) d θ = − 2 b θ m = − 4 π . (33)
This is the total change of the slope of the “outer solution” as the turning point is crossed.
This Section focuses on the case of a polynomial V(x):
x ¨ ( t ) + x ( t ) + ∑ n = 1 N a 2 n + 1 x ( t ) 2 n + 1 = 0 , a 2 N + 1 > 0 . (34)
When all coefficients, a2n+1, in Equation (34), are constant, the leading power, x2N+1, dominates around the turning points in the limit of very high energy, E, as it grows like a2N+1. Still, the nonlinearity in Equation (34) does not obey conditions (10), (11) and (16) for fixed (2N + 1). The asymptotic saw-tooth profile of the solution emerges only in the double limit E → ∞ and N → ∞. As this case has been discussed in detail in [
μ = 1 / ( ( a 2 N + 1 ) 1 / 2 N a ) ≪ 1 , (35)
the period of oscillation, T, is found to be O(μN):
T = μ N G ( N , μ ) , G = 4 N + 1 ( 1 + O ( 1 / N ) ) + O ( μ 2 p ) , (36)
where (2p + 1) is the power of the next-to the leading term.
Under the scaling of Equation (4)), Equation (34) becomes:
η ″ ( θ ) + ( G 2 π ) 2 { μ 2 N η ( θ ) + ∑ n = 1 N a 2 n + 1 ( a 2 N + 1 ) n / N μ 2 ( N − n ) μ η ( θ ) 2 n + 1 } = 0 . (37)
In the double limit of a → ∞ (μ → 0) and N → ∞, for all |η(θ)| < 1, Equation (37) is reduced to
a → ∞ : η ″ ( θ ) + ( G / ( 2 π ) ) 2 η ( θ ) 2 N + 1 = 0 ⇒ N → ∞ | η ( θ ) < 1 | η ″ ( θ ) = 0 , (38)
leading to the emergence of the “outer solution”.
The boundary-layer-like analysis yields:
η outer ( θ ) = ( 2 / π ) / ( ( π / 2 ) − θ ) ( 1 + O ( 1 / ( N + 1 ) ) ) η inner ( θ ) = 1 − 2 ( N + 1 ) π 2 θ 2 ( 1 + O ( 1 / ( N + 1 ) ) ) θ m → a → ∞ π 2 N + 2 ( 1 + O ( 1 / ( N + 1 ) ) ) . (39)
Nonlinear violence emerges in the double limit of a → ∞ and N → ∞, leading to the same periodic saw-tooth profile for η(θ) as in previously discussed systems.
δ-function profile for acceleration
As in Sections 2.1.1 and 2.1.2, η''(θ) tends to a periodic sequence of δ-functions (Equation (18)). Again, the jump in η'(θ) generated by the δ-function is well approximated by cumulative change of η'(θ) over “ηinner” of Equation (39):
η ′ ( θ m ) − η ′ ( − θ m ) ≈ ∫ − θ m + θ m η ″ inner ( θ ) d θ → a → ∞ − 4 π ( 1 + O ( 1 / ( N + 1 ) ) ) → N → ∞ − 4 π (40)
In the extreme relativistic limit of the relativistic harmonic oscillator,
m ω 0 2 x + m x ¨ ( 1 − ( x ˙ / c ) 2 ) 3 / 2 = 0 . (41)
x(t) tends to a periodic saw-tooth profile. Over the years, this has been demonstrated through numerical solutions of the Equation (41) [
To present numerical solutions of Equation (41) it pays to use dimensionless variables:
t ˜ = ω 0 t , x ˜ ( t ˜ ) = ( ω 0 / c ) x ( t ) . (42)
Equation (41) is transformed into
x ˜ ( t ˜ ) + x ˜ ¨ ( t ˜ ) ( 1 − ( x ˜ ˙ ( t ˜ ) ) 2 ) 3 / 2 (43)
Define
β = max ( | x ˙ ( t ) | / c ) = max ( x ˜ ˙ ( t ˜ ) ) . (44)
An example of phase-space plots for x ˜ ˙ ( t ˜ ) vs. x ˜ ( t ˜ ) is shown in
That the solution of Equation (41) tends in the extreme relativistic limit to the same limiting saw-tooth profile as in previous Sections may not be surprising, as x ˙ ( t ) tends to ±c, the speed of light, throughout most of the period of oscillations. The characteristic that is common to the present system and the previously discussed systems, is that the solution of Equation (41) also exhibits a boundary-layer-like structure: most of the time, the solution is well approximated by the linear time dependence of an “outer solution”. Around each turning point, the solution of x(t) has a strongly curved profile over a very small fraction of the total period, T, a fraction that tends to zero as E → ∞. The curved part of the solution is very well approximated by an “inner solution”. Dominance of the nonlinearity near the turning points ensures that the asymptotic form of the solution of Equation (41) is identical to that found in Sections 2.1.1 - 2.1.3. This happens despite the fact that there are qualitative differences between the relativistic harmonic oscillator and the previously discussed systems. In the previous systems, one has:
E → ∞ : a ( E ) → ∞ , max | x ˙ ( t ) | ∝ E → ∞ , T → 0 , (45)
where E is the total energy. In the case of the relativistic harmonic oscillator, one has:
E → ∞ : a ( E ) → ∞ , max | x ′ ( t ) | → c , T → ∞ . (46)
Moreover, in the systems discussed in previous subsections, nonlinear violence emerged without resorting to the transformation of the dynamical equations into traditional boundary-layer systems of the form of Equation (2). In the case of Equation (41), nonlinear violence emerges in the extreme relativistic limit, through such a transformation.
Energy conservation for Equation (41) yields:
a ( E ) = max | x | = 2 c ω 0 1 − 1 − β 2 ( 1 − β 2 ) 1 / 4 , max | x ˙ | = β c . (47)
The transformation of Equation (4) yields a periodic boundary value problem:
η ( θ ) + ( 2 π ω 0 T ) 2 η ″ ( θ ) ( 1 − ( 2 π x max c T ) 2 η ′ ( θ ) 2 ) 3 / 2 = 0 , η ( 0 ) = − η ( π ) = η ( 2 π ) = 1 η ( π / 2 ) = 0 . (48)
η(θ) is (2π)-periodic in θ and bounded: |η(θ)| ≤ 1. The period, T, is given by [
T = 1 ω 0 4 2 ( 1 + 1 1 − β 2 E ( k ) − ( 1 / 1 + 1 1 − β 2 ) K ( k ) ) k 2 = 1 − 1 − β 2 1 + 1 − β 2 , (49)
where E(k) and K(k) are the complete elliptic integrals [
E ( k ) = ∫ 0 π / 2 1 − k 2 sin φ 2 d φ , K ( k ) = ∫ 0 π / 2 ( 1 / 1 − k 2 sin φ 2 ) d φ . (50)
Applying energy conservation to Equation (48) and exploiting properties of the elliptic integrals [
η max ≡ max ( | η | ) = 1 , η ′ max ≡ max | ( η ′ ( θ ) ) | = β c T 2 π x max → β → 1 2 π . (51)
To measure the proximity to the extreme relativistic limit, define a small parameter, μ:
β 2 = 1 − μ 4 , ( μ ≪ 1 ) . (52)
Expansion in power of μ, (for elliptic integrals, see [
η ( θ ) + π 2 8 μ 2 ( 1 + O ( μ 4 log μ ) ) η ″ ( θ ) ( 1 − π 2 4 ( 1 − μ 2 + O ( μ 4 ) ) η ′ ( θ ) 2 ) 3 / 2 = 0 η ( 0 ) = 1 , η ′ ( 0 ) = 0 . (53)
The system exhibits the characteristics of a boundary layer problem (see Equation (2)): the highest derivative in the Equation (53) is multiplied by a small parameter.
To find the inner solution corresponding to Equation (53) near the turning point, θ = 0, (η(0) = 1, η'(0) = 0), observe that η' is very close to zero there. Replacing η' by zero in Equation (53), the O(μ2) approximation to Equation (53) becomes:
η ( θ ) + π 2 8 μ 2 η ″ ( θ ) = 0 . (54)
For the boundary conditions at the turning point, Equation (54) is solved by
η inner ( θ ) = cos ( 2 2 π θ μ ) . (55)
To obtain the “outer solution” we focus on the vicinity of θ = (π/2). As the chosen turning point is at θ = 0, two observations hold near θ = (π/2): η(θ) ≈ 0 and η' ≈ ( − η ′ max ). Substituting η(θ) = 0 in Equation (53), yields the approximate equation;
η ″ = 0 ⇒ η outer = η ′ max ( π 2 − θ ) . (56)
Exploiting the leading-order term in η ′ max by expanding Equation (51) in powers of μ, yields:
η ′ max ≡ max | ( η ′ ( θ ) ) | = 2 π + μ 2 π + O ( μ 4 log μ ) . (57)
Using Equations (55) - (57), an O(μ2) approximation for the matching point, θ = θm, at which the slope of ηinner becomes equal to the constant slope of the ηouter, which prevails throughout most of the range 0 ≤ θ ≤ (π/2) one finds:
η ′ inner ( θ m ) ≈ − η ′ max ⇒ θ m ≈ π 4 μ 2 ( 1 + 1 2 μ 2 ) . (58)
Requiring that ηinner and ηouter coincide at θ = θm yields:
η outer ( θ ) = 2 π ( π 2 − θ ) + μ 2 4 ( 1 − 8 π 2 θ ) + O ( μ 4 ) . (59)
δ-function profile for acceleration As in previous sections, η''(θ) tends to a periodic sequence of δ-functions with alternating signs (Equation (18)). Again, this is well approximated by the cumulative change of η'(θ) over “the inner solution” as the turning point is crossed. Using Equations (55) and (58), one obtains the total change in the slope of the “outer solution” as the turning point is crossed:
η ′ ( θ m ) − η ′ ( − θ m ) ≈ ∫ − θ m + θ m η ″ i n n e r ( θ ) d θ = − 4 2 π μ sin ( μ ( 2 + μ 2 ) / ( 2 2 ) ) = − 4 π ( 1 + 5 12 μ 2 + O ( μ 4 ) ) → μ → 0 − 4 π . (60)
In the extreme relativistic limit (μ → 0), the preceding analysis ensures that the linear “outer solution” describes the solution very well and the time interval around the turning points, where the solution is described by xinner tends to zero. As a result, the asymptotic profile of the solution is also a saw-tooth one, as in Section 2.1.
A new vista opens up when coefficients of the non-leading monomials in Equation (34) grow with energy, i.e., if for some p < N,
a 2 p + 1 → a ( E ) → ∞ α a ( E ) 2 ( N − p ) ⇒ a 2 p + 1 x 2 p + 1 / a 2 N + 1 a ( E ) 2 N + 1 → x → a ( E ) a 2 p + 1 a 2 p + 1 / a 2 N + 1 a ( E ) 2 N + 1 → O ( 1 ) , (61)
if, in addition, a2p+1 > 0 and a2N+1 < 0. More than one term in the polynomial participates in nonlinear violence, and owing to the opposite signs, they compete with one another. For a → ∞, an additional leading monomial shows up in Equation (38):
η ″ ( θ ) + ( G 2 π ) 2 { α η ( θ ) 2 p + 1 − η ( θ ) 2 N + 1 } = 0 . (62)
Unlike the case of constant coefficients in a polynomial potential (Section 2.1.3), the additional requirement of N → ∞ is not necessary here. The characteristics of the solutions for different values of p and N are similar, provided α = 1. Hence, one example of physical significance is discussed, the Duffing equation with an energy-dependent liner term:
x ¨ ( t ) + ( 1 + 1 2 a ( E ) 2 ) x − 1 2 x ( t ) 3 . (63)
Equation (63) was derived by an elegant transformation of the equation of motion of the relativistic harmonic oscillator, Equation (41), using proper time (time as measured on the moving mass rather than time measured by an external observer) and scaled, dimensionless coordinates [
η ″ ( θ ) + T 2 4 π 2 η ( θ ) + T 2 4 π 2 1 2 a 2 ( η ( θ ) − η ( θ ) 3 ) = 0 , ( η ( 0 ) = 1 , η ′ ( 0 ) = 0 ) . (64)
The period T is given by [
T = 4 E ( i a 2 ) = 8 a 2 + 4 E ( a a 2 + 4 ) = 8 log ( 2 a ) a − 8 log ( 2 a ) − 1 a 3 + ⋯ , (65)
where properties of E(k) (see Equation (50)) are found in [
0 = 1 ( a T ) 2 η ″ ( θ ) + 1 4 π 2 a 2 η ( θ ) + 1 8 π 2 ( η ( θ ) − η ( θ ) 3 ) ≈ 1 ( 8 log ( 2 a ) ) 2 η ″ ( θ ) + 1 4 π 2 a 2 η ( θ ) + 1 8 π 2 ( η ( θ ) − η ( θ ) 3 ) , η ( 2 π + θ ) = η ( θ ) (66)
The highest derivative is multiplied by a small parameter, which varies slowly as a → ∞.
Equation (66) has a closed-form solution [
η ( θ ) = s n [ a 2 + 4 2 T ( θ + π / 2 ) 2 π , a a 2 + 4 ] . (67)
In Equation (67), sn is a Jacobi elliptic function [
Applying energy conservation to Equation (66), or computing the derivative of η'(θ = π/2) through Equation (67), one obtains the maximal value of the derivative to be:
η ′ max = 4 + a 2 4 π T = 2 π log ( 2 a ) + 2 a 2 ( 1 + log ( 2 a ) ) . (68)
The “inner solution” describes the rapid change around θ = π/2. It is obtained by solving for η(θ) in a power series in (π/2 − θ): Through O((π/2 − θ)2), it is given by:
η inner ( θ ) = η ′ max ( π 2 − θ ) + A ( π 2 − θ ) 2 . (69)
The “outer solution”, describing the moderate change around θ = 0 is found by solving Equation (66) around η(θ) = 1:
η outer ( θ ) = 1 − δ ( θ ) (70)
Substituting Equation (70) in Equation (66), the linearized equation for δ ( θ ) ≪ 1 becomes:
T 2 4 π 2 { 1 − ( a 2 − 1 ) δ ( θ ) } + δ ″ ( θ ) = 0 , δ ( 0 ) = δ ′ ( 0 ) = 0 . (71)
Equation (71) is solved by
δ ( θ ) = 2 sinh ( a 2 − 1 T θ 4 π ) 2 / ( a 2 − 1 ) ≈ 2 E ( i A 2 ) π 2 θ 2 . (72)
The matching point, θm, is found by requiring
η inner ( θ m ) = η outer ( θ m ) , η ′ inner ( θ m ) = η ′ outer ( θ m ) . (73)
One finds:
θ m = π 2 − π 2 log ( 2 a ) + 2 π log ( 2 a ) 2 − log ( 2 a ) + 1 a 2 + ⋯ A = η ′ max 2 ( 1 4 + log ( 2 a ) 2 − 4 log ( 2 a ) 2 a 2 ) . (74)
The rate of approach as a → ∞ is slow. The matching point, θm, deviates from (π/2) by O(1/log2a).
If V(x) is not symmetric under reflection in x, the amplitudes at the two turning points are not equal up to a sign. Consequently, the analysis has to be performed separately close to any turning point, around which nonlinear violence prevails. The overall scaling of the solution (x = a ∙ η) in Equation (4) has to be replaced near each turning point by (x = xtur(E)×η), where xtur(E) is the amplitude at that turning point. Finally, while the transformation (t → θ) of Equation (4) yields
(2π)-periodicity in θ, there is no symmetry between (0 ≤ θ ≤ π) and (π ≤ θ ≤ 2π). For example, if only one turning point exhibits nonlinear violence, T is a sum of two terms: a vanishingly small interval of time (spent around that turning point affected by nonlinear violence) and the rest of the time, spent over most of the oscillation.
Except for technical details, the analysis, specifically, the procedure for matching the “inner” and “outer” solutions, is the same as presented in Sections 2.1 - 2.4. Hence, instead of analyzing a general system, the analysis is carried out in Sections 3.2 - 3.3 for a specific example, where the solution is affected by nonlinear violence only around one turning point.
Consider a harmonic oscillator with an exponential nonlinearity:
x ¨ ( t ) + x ( t ) + e x ( t ) = 0 . (75)
The potential energy is asymmetric:
V ( x ) = 1 2 x 2 + e x ⇒ V ( x ) → x → − ∞ 1 2 x 2 , V ( x ) → x → + ∞ e x E = 1 2 x max ( E ) 2 + e x max ( E ) . (76)
For large E, at large negative x, V(x) of Equation (76) tends to that of a harmonic oscillator. Hence, the profile of oscillations is expected to be close to a trigonometric function. Nonlinear violence occurs only around the turning point, xmax(E) > 0, as there the potential energy grows rapidly. (xmax(E) → ∞ as E → ∞.) If the initial conditions are such the x(0) is that turning point, so that x ˙ ( 0 ) = 0, turning points occur at t0 = 0, T, 2T, …
As E → ∞, for x(t) < 0, the exponential term in Equation (75) becomes rapidly negligible. Hence, the time spent over the x(t) < 0 branches of oscillations approaches π (half of a full sinusoidal oscillation). The time spent around the turning point, which is governed by nonlinear violence, is vanishingly small. Hence, T tends to π as E → ∞. For x(0) = 10, numerical integration yields T = 3.25.
Numerical ExamplesThe boundary-layer structure that emerges around xmax(E) is demonstrated
To find an approximation for the solution of Equation (75) near the turning point, a = xmax(E) > 0, for large E, write x(t) as:
x ( t ) = a η ( τ ) , τ = α t , 0 ≤ η ≤ 1 . (77)
Equation (75) becomes
1 a e a η ( τ ) + η ( τ ) + α 2 η ″ ( τ ) = 0 . (78)
Energy conservation for Equation (75) yields
a = E ≫ 1 log E − ( log E ) 2 2 E + o ( ( log E ) 2 2 E ) . (79)
To find α, substitute Equation (77) in Equation (75) and approximate the “inner solution” by:
η inner ( τ ) = 1 − τ 2 + O ( τ 4 ) . (80)
To leading order, one obtains
α = E 2 log E ( 1 − ( log E ) 2 4 E + 3 log E 4 E + o ( 1 E ) ) . (81)
As the solution evolves into a trigonometric profile once away from the turning point, we expect the “outer solution” to provide a good approximation (linear in time) for the full solution for a limited range in time. For this reason, scaling of the time variable with α of Equation (81), and a of Equation (79) will be employed also in the generation of the “outer solution”. Equation (75) becomes:
2 ( log E E + E η ( τ ) − 1 ) ( 1 + O ( 1 / E ) ) η ( τ ) + η ″ ( τ ) = 0 . (82)
For E → ∞ and (1 − |η(τ)|) = O(1)) > 0, Equation (82) tends to:
η ″ ( τ ) = 0 . (83)
This yields the “outer solution”:
η outer ( τ ) = A + B τ . (84)
Matching of the “inner” and “outer” solutions by making their values and their derivatives equal at t = tm yields the matching point:
t m = 2 E ( 1 + log E 2 E + o ( 1 E ) ) . (85)
For a turning point at t = t0, the “inner solution” is found to be
x ( t ) inner ≈ a ( 1 − E ( t − t 0 ) 2 2 log E + E 2 ( t − t 0 ) 4 24 log E ) + O ( E 3 ( t − t 0 ) 6 ) , (86)
and the outer solution:
x ( t ) outer ≈ a ( ( 1 + 7 6 log E + O ( 1 E ) ) + 2 E log E ( 1 + log E 2 E − O ( 1 E ) ) | t − t 0 | ) . (87)
A comparison of the approximations provided by the “inner solution” and the “outer solution” with the numerical solution of Equation (75) around the turning point amplitude, x(t0) = xmax = 10 and t0 = 3.25, is shown in
Velocity change across turning point Using Equations (83) and (84), the cumulative change of η'(t) over the diminishingly small interval, (t0 − tm, t0 + θm,), (θm → 0 for E → ∞) is found to coincide with the change in the slope of the “outer solution”, xout, as one crosses the turning point from left to right:
x ′ ( t 0 + t m ) − x ′ ( t 0 − t m ) = ∫ t 0 − t m t 0 + t m x ″ i n n e r ( t ) d t = − 2 2 E log E ( 1 + O ( 1 / E ) ) , (88)
again, asymptotically yielding the overall change in x'(t) as the turning point is crossed.
At high E, the motion of a particle in a central field in two space dimensions offers new interesting boundary-layer phenomena if the angular momentum is not equal to zero. The evolution of the radius, r, of a unit mass particle is determined by [
r ″ + ∂ r V e f f ( r ) = 0 V e f f ( r ) = l 2 / ( 2 r 2 ) + V ( r ) . (89)
Here V(r) is the potential and l is the conserved angular momentum:
l = r 2 ( t ) φ ˙ ( t ) = C o n s t . (90)
φ is the angle of rotation of the moving mass around the origin.
For E < 0, the solution classical motion under a gravitational potential,
V ( r ) = − 1 / r , (91)
is bounded and periodic. For E > 0, the solution visits the point of the minimal radius, rmin(E), once and wonders off to infinity [
Energy conservation for Equation (89) with V(r) of Equation (91),
E = l 2 / ( 2 r ( t ) 2 ) − 1 / r ( t ) + 1 2 r ˙ ( t ) 2 = l 2 / ( 2 r min ( E ) 2 ) − 1 / r min ( E ) , (92)
yields:
r min ( E ) = 2 l 2 E + 1 − 1 2 E = l 2 E − 1 2 E + O ( E − 3 / 2 ) (93)
and
r ˙ ( t ) = ± 2 E + ( 2 / r ( t ) ) − ( l / r ( t ) ) 2 → r ( t ) → ∞ ± 2 E (94)
Hence, the “outer solution” is given by:
r outer ( t ) = A + 2 E t (95)
For the “inner solution”, as rmin is a local minimum around t = 0, write:
r inner ( t ) = r min ( 1 + α t 2 ) . (96)
Substituting Equation (96) in Equation (89) yields:
α = l 2 2 r min ( E ) 4 ( 1 − ( r min ( E ) / l 2 ) + O ( r min ( E ) 2 ) ) . (97)
Using Equations (95) - (97), matching the “inner” and “outer” solutions and their time derivatives at the matching time, tm, yields:
t m = 2 E r min ( E ) 3 l 2 − r min ( E ) = l 2 E ( 1 + O ( 1 E ) ) , A = l 2 2 2 E l 2 + 1 = l 2 2 E ( 1 + O ( 1 / E ) ) . (98)
Thus, the matching point, tm, vanishes like E-1.
Velocity change across turning point, rmin Using Equations (90) and (93) - (98), the total change in the velocity r ˙ ( t ) , around rmin over the time interval −tm ≤ t ≤ + tm, computed in terms of rinner, reproduces the full change in the slope of the “outer solution” from negative to positive linear slopes:
r ˙ ( + t m ) − r ˙ ( − t m ) ≈ ∫ − t m + t m r ¨ inner ( t ) d t = 2 r min ( E ) α t m = 2 2 E ( 1 + O ( 1 E ) ) . (99)
Rapid rotation at turning point, rmin
The last interesting aspect is the variation of the angle of rotation, φ, in time as E → ∞. Close to the turning point, Equations (90), (94), (96) and (97) yield:
φ ˙ ( t → 0 ) = l r min ( E ) 2 = 2 E l ( 1 + ( 2 / E ) / l + O ( 1 / E ) ) φ ˙ ( t → t m ) = l r ( t m ) 2 = 8 E 9 l ( 1 + ( 4 / 3 ) ( 2 / E ) / l + O ( 1 / E ) ) . (100)
Thus, φ ˙ ( t ) becomes infinite in the vicinity of the turning point as E → ∞. However, soon the “outer solution” takes over, for which:
φ ˙ ( t ≫ t m ) = l r outer ( t ) 2 ≈ 1 2 E t 2 → E → ∞ 0 . (101)
In summary, close to rmin, the angle, φ, varies rapidly between two constants over a vanishingly small time interval around t = 0. Away from that interval it varies very slowly with time up to the large turning point, rmax. In the limit E → ∞, φ exhibits a finite jump around the rmin. This point will be expanded in Section (4.2).
Consider in Equation (89) V(r), which grows rapidly at large r. To be specific choose:
V ( r ) = e r . (102)
The motion is oscillatory as the total energy is:
E = ( l 2 / ( 2 r 2 ) ) + e r . (103)
Nonlinear violence, leading to a boundary-layer phenomenon, emerges around the turning point at the maximal radius, rmax. For E ≫ 1 :
r max ≈ log E . (104)
Nonlinear violence emerges also in the vicinity of rmin if l ≠ 0 and the contribution of the angular momentum term dominates:
l 2 / ( 2 r min 2 ) E → E → ∞ 1 ⇒ V ( r min ) l 2 / ( 2 r min 2 ) → E → ∞ 0 . (105)
Equation (105) holds for V(r) of Equation (102). The analysis is identical to that presented in Section 4.1, hence, not repeated here.
Numerical examples are shown for E = 106 and l = 1.
Energy conservation for Equation (89) with V(r) of Equation (102),
E = l 2 / ( 2 r ( t ) 2 ) + e r ( t ) + 1 2 r ˙ ( t ) 2 = l 2 / ( 2 r max ( E ) 2 ) + e r max , (106)
yielding Equation (104) for rmax and
r ˙ ( t ) = ± 2 E − 2 e r ( t ) − ( l / r ( t ) ) 2 → E → ∞ r ( t ) ≪ r max ± 2 E . (107)
Hence, the “outer solution” is given by:
r outer ( t ) = A + 2 E t . (108)
For the “inner solution”, as rmax is a local minimum around t = 0, write:
r inner ( t ) = r max ( 1 − α t 2 ) . (109)
Substituting Equation (109) in Equation (89) with the potential of Equation (102) yields:
α = e r max 2 r max ( 1 − l 2 e − r max / r max 3 ) ≈ E 2 log E ( 1 − l 2 / ( E ( log E ) 3 ) ) . (110)
Using Equations (108)-(110), matching the “inner” and “outer” solutions and their time derivatives at the matching time, tm, yields:
t m = 2 e − r max / 2 1 − ( e − r max l 2 / r max 3 ) = 2 E / 12 ( 1 + O ( 1 E ) ) A = ( 1 + r max ) ( 1 + O ( e − r max l 2 r max 3 ) ) ≈ 1 + log E ( 1 + O ( l 2 E ( log E ) 3 ) ) . (111)
Thus, the matching point, tm, vanishes as E-1/2.
Velocity change across turning point, rmax Using Equations (108) and (111), the total change in the velocity r ˙ ( t ) around rmax as one crosses the turning point, is well reproduced by the contribution of rinner:
r ˙ ( + t m ) − r ˙ ( − t m ) = ∫ − t m + t m r ¨ inner ( t ) d t = − 2 2 e r max ≈ − 2 2 E . (112)
Again, this is the total change in r ˙ outer ( t ) .
Near rmin
The angular velocity, φ ˙ ( t ) , is shown in
Δ φ = ∫ − t m + t m ( L / r inner ( t ) 2 ) d t = 2 3 + 2 tan ( 1 / 2 ) = 1.53709 (113)
If instead of rinner we use the numerical solution of Equation (89) for r(t), with l = 1, the potential of Equation (102), and E = 106 in Equation (113) the computed jump obtained is 1.5715. The exact result is that, as E → ∞, Δφ jumps across rmin by (π/2) = 1.5708 [
Near rmax
Assuming that rmax occurs at t = 0, one finds in the vicinity of rmax:
r inner ( t ) = r max ( 1 − ( e r max / ( 2 r max ) ) ( 1 + O ( ( e − r max / r max 3 ) ) ) t 2 ) r outer = A + 2 E t , A = r max + 1 + O ( e − r max ( 1 / r max 2 ) ) t m = 2 e − r max / 2 ( 1 + O ( ( e − r max / r max 2 ) ) ) . (114)
The integral of Equation (113) for rinner of Equation (114) reveals that the change in φ(t) across rmax in the time interval (−tm, + tm), with tm of Equation (114) vanishes as E → ∞:
Δ φ = O ( e − r max / 2 / r max 2 ) = O ( 1 / ( E 1 / 2 ( log E ) 2 ) ) , (115)
In summary, in agreement with what the numerical solution shows, φ(t) varies rapidly (a jump of (π/2) in the limit E → ∞) when crossing rmin and by a vanishingly small change when crossing rmax. Outside the short time intervals spent around rmin (Equation (93)) and rmax (Equation (104)), the total variation of φ ( t ) along each branch of the oscillation cycle is (3/4π), occurring at a very low rate, O(1/(E ∙ t2)).
Comment: Choice of the exponential potential of Equation (102), the contribution of which to Veff of Equation (89) becomes negligible at small r, was motivated by the role played by the angular momentum in generating rapid rotation around rmin. A detailed analysis (not presented here) shows that, if the exponential potential of Equation (89) is replaced by V(r) such that its contribution to Veff of Equation (89) dominates also at small r, then, depending on the details of V(r), rapid rotation around rmin may or may not emerge.
Nonlinear oscillations with high oscillation energy are of relevance in many areas of science and technology: communications science, electrical circuits, electronic devices, biological systems and more. A recent example has emerged in low-temperature physics, in the oscillations of ions in a Bose-Einstein condensate [
There is a large family of oscillatory systems whose solutions are affected by nonlinear violence, leading to boundary layer-like characteristics when the oscillation energy is extremely high. These characteristics emerge independently of whether the dynamical equation of the oscillator can or cannot be cast in the traditional form of boundary-layer problems.
Planar motion of a particle in a central field displays a new unique phenomenon, which opens the door to interesting behavior of oscillators in higher space dimensions. Boundary-layer characteristics emerge around the turning point of the largest radius owing to the growth of the potential energy. If the angular momentum, l, does not vanish, nonlinear violence may emerge around the turning point of the smallest radius, rmin. The latter is accompanied by a boundary-layer characteristic behavior of the angle that determines the position of the rotating particle in the plane. At large overall energy, the change in the angle when crossing rmin is very rapid and tends to a discrete jump of (π/2) as E → ∞.
In all the cases discussed here, the change of velocity and of rotation angle when a turning point is crossed are well approximated by computing them with the relevant “inner solution” over the time interval: (−tm ≤ t ≤ +tm).
In the case of (1 + 1)-dimensional energy conserving nonlinear oscillatory systems, the effect of nonlinear violence depends on the detailed structure of the equations of motion. If the potential energy of the system is not symmetric under space reflection, nonlinear violence may emerge around one or both turning points. If the potential energy is symmetric under reflection in x, these characteristics emerge at both turning points. The solution for x(t) then tends to a simple periodic profile: examples of saw-tooth and step-function profile were discussed.
The analysis discussed in this paper leads to the conclusion that, as E → ∞, the asymptotic profiles of solutions of a large family of oscillatory systems tend to similar standard forms, such as saw-tooth or step-function profiles. Hence, if such profiles are observed experimentally in oscillating physical systems, it may be difficult to decipher the equations of motion. A possible future research direction that will improve our understanding of high-energy nonlinear oscillations is the development of an expansion analysis around the asymptotic (saw-tooth or step-function) profiles, for the deviations of the actual solution from the asymptotic profile. This may be very important in deciphering the equations of motion governing the dynamical system, for which only experimental observations exist and theory has not been developed yet.
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Zarmi, Y. (2024) Nonlinear Violence in Nonlinear Oscillations at High Energy. Applied Mathematics, 15, 65-95. https://doi.org/10.4236/am.2024.151007