Boundary-layer features in conservative nonlinear oscillations

Zarmi, Yair

doi:10.1007/s11071-022-07599-w

Cited by 1 publication

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“…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 [36], the Duffing [37] [38] [39] [40], quintic [41] [42] [43] [44] and relativistic harmonic [45] [46] [47] oscillators), or can be computed to any desired level of accuracy. This idea has been exploited in the analysis of the case of a polynomial nonlinearity when both the maximal oscillation amplitude, a(E), and the highest power in the polynomial, (2N + 1), both tend to infinity [48].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Violence in Nonlinear Oscillations at High Energy

Zarmi

2024

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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.

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Section: Introductionmentioning

confidence: 99%

Nonlinear Violence in Nonlinear Oscillations at High Energy

Zarmi

2024

Self Cite

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Boundary-layer features in conservative nonlinear oscillations

Cited by 1 publication

References 32 publications

Nonlinear Violence in Nonlinear Oscillations at High Energy

Nonlinear Violence in Nonlinear Oscillations at High Energy

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