On Near Integrability of Some Impact Systems

Pnueli, Michal; Rom‐Kedar, Vered

doi:10.1137/18m1177937

Cited by 8 publications

(43 citation statements)

References 31 publications

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“…) is a sum of two potentials each depending on one variable, By property 4, S3B Hamiltonians have bounded energy surfaces. For simplicity of presentation, since all the analysis here is local near a regular torus of (1), assume hereafter that each of the potentials V i (q i ), i = 1, 2 has a single, simple minimum, located at q ic = 0 such that V i (q ic ) = 0 (and so min H int = 0), and are convex, so in particular q i • V i (q i ) > 0 for q i = 0 (for potentials with multiple number of extremal points, our results apply to the behavior near each regular torus of the smooth system belonging to a branch of the Liouville foliation of isoenergy surfaces, see [23]).…”

Section: Setupmentioning

confidence: 83%

“…The second term in (1), V c , represents a small, regular perturbation by coupling and is assumed to be C r+1 smooth. Thus V c (•) is bounded on the bounded energy surfaces (see [23]), and the bound generally depends on E, the energy level. Hereafter we assume that E = O(1).…”

Section: Setupmentioning

confidence: 99%

“…where b is either a fixed large number (representing an impassable wall for all energies of interest -can also be taken as infinity) or zero (representing the smooth Hamiltonian system without the impact). More generally, it is possible to extend the analysis and consider a C r+1 smooth perturbation of the wall as in [23] and a steep smooth potential instead of a wall as in [11,23], however, for now, we present the near tangent analysis for the case of a single perpendicular wall and leave such extensions to future studies. Summarizing, for b > 0 we consider the C r -smooth near-integrable Hamiltonian flow in the half plane q 1 > q w 1 with elastic impacts at the vertical wall boundary: Definition 2.2.…”

Section: Setupmentioning

confidence: 99%

“…Since the wall is vertical and respects the separability symmetry of the underlying integrable Hamiltonian flow, by the rule of elastic reflection and the symmetry of the kinetic energy term, the partial energies (E 1 , E 2 ) are preserved upon impact and the motion remains rotational on level sets. Hence, the unperturbed wall system ( = 0 in (1)), is Liouville integrable [23,24]. This system has singular tangent level sets (E 1 , E 2 ) exactly when E 1 = V 1 (q w 1 ) (then p 1 = 0 at the wall), and these level sets correspond to a torus in the allowed region of motion for all E 2 > 0 if and only if q w 1 < 0.…”

Section: Setupmentioning

confidence: 99%

“…section is also transverse to unperturbed tori that are bounded away from the tangent torus, and since under small conservative perturbation most of these tori persist [23], a finite neighborhood of the tangent torus remains invariant, separating between perturbed impacting and non-impacting regimes. We call this region the tangency band.…”

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confidence: 99%

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Near tangent dynamics in a class of Hamiltonian impact systems

Pnueli¹,

Rom-Kedar²

2021

Preprint

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Tangencies correspond to singularities of impact systems, separating between impacting and non-impacting trajectory segments. The closure of their orbits constitute the singularity set, which, even in the simpler billiard limit, is known to have a complex structure. The properties of this set are studied in a class of near integrable two degrees-of-freedom Hamiltonian impact systems. For this class of systems, in the integrable limit, on iso-energy surfaces, tangency appears at an isolated torus. We construct a piecewise smooth iso-energy return map for the perturbed flow near such a tangent torus. Away from the singularity set, this map has invariant curves, so, the singularity set is included in a limiting singularity band. An asymptotic upper bound of this band width is found for both non-resonant and resonant tangent tori.In the Diophantine case the band extent to both the impacting and non-impacting regimes is, asymptotically, of order . In the resonance case its asymptotic extent to the non-impacting regime is at most of order √ whereas its maximal asymptotic extent to the impacting regime is of order 2/3 . Studying numerically a truncated standard form of the return map which is integrable at = 0 and has a square-root singularity with a singularity parameter α, reveals that only the strongest resonance obeys this maximal extent resonance scaling. We find that for all but one resonant case the singularity band has connecting orbits from its lower to its upper boundaries. The number of iterates required for an orbit to visit close to both the lower and upper boundaries of the singularity band diverges as α 2 is decreased, giving rise to multitude of transient complex phenomena. On the other hand, large α values at a fixed small lead to a significant reduction in this connection time.

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Section: Setupmentioning

confidence: 83%

Section: Setupmentioning

confidence: 99%

Section: Setupmentioning

confidence: 99%

Section: Setupmentioning

confidence: 99%

mentioning

confidence: 99%

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Near tangent dynamics in a class of Hamiltonian impact systems

Pnueli¹,

Rom-Kedar²

2021

Preprint

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Impact Hamiltonian systems and polygonal billiards

Becker,

Elliott,

Firester

et al. 2024

Hamiltonian Systems

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Non-uniform ergodic properties of Hamiltonian flows with impacts

Frączek

Rom‐Kedar²

2021

Ergod. Th. Dynam. Sys.

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The ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

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On Near Integrability of Some Impact Systems

Cited by 8 publications

References 31 publications

Near tangent dynamics in a class of Hamiltonian impact systems

Near tangent dynamics in a class of Hamiltonian impact systems

Impact Hamiltonian systems and polygonal billiards

Non-uniform ergodic properties of Hamiltonian flows with impacts

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