Parabolic resonances in near integrable Hamiltonian systems

Litvak-Hinenzon, Anna; Rom‐Kedar, Vered

doi:10.1063/1.1302407

Cited by 4 publications

(20 citation statements)

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“…This observation falls into a more general framework which distinguishes between different types of PR according to the relation between the actions which govern the bifurcating tori and the resonance directions. While the persistence theorems apply equally well to all cases, the corresponding perturbed orbits exhibit significantly different behaviours as demonstrated in [29,32] for the two possible types of 1-PR in 3 DOF systems.…”

Section: Iso-energetic Prs In a 5 Dof Modelmentioning

confidence: 70%

“…. , 7, are inserted for generality and compatibility with the 3 DOF phenomenological model which was studied in [31,29,32] (see equation (3.23)). We show below that for any fixed non-zero value of the external parameters there exists an energy surface on which an infinitesimal family of 1-resonant elliptic tori, emanating from a 1-resonant parabolic torus in the direction of I 1 , resides.…”

Section: Degenerate and Iso-energetic Resonances In A 4 Dof Modelmentioning

confidence: 99%

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Resonant tori and instabilities in Hamiltonian systems

Litvak-Hinenzon

Rom‐Kedar

2002

Nonlinearity

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The existence of lower-dimensional resonant bifurcating tori of parabolic, hyperbolic and elliptic normal stability types is proved to be generic and persistent in a class of n degrees of freedom (DOF) integrable Hamiltonian systems with n 3. Parabolic resonance (PR) (respectively, hyperbolic or elliptic resonance) is created when a small Hamiltonian perturbation is added to an integrable Hamiltonian system possessing a resonant torus of the corresponding normal stability. It is numerically demonstrated that PRs cause intricate behaviour and large instabilities. The role of lower-dimensional bifurcating resonant tori in creation of instability mechanisms is illustrated using phenomenological models of near integrable Hamiltonian systems with 3, 4 and 5 DOF. Critical n values for which the system first persistently possesses mechanisms for large instabilities of a certain type are found. Initial numerical studies of the rate and time of development of the most significant instabilities are presented.

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Section: Iso-energetic Prs In a 5 Dof Modelmentioning

confidence: 70%

Section: Degenerate and Iso-energetic Resonances In A 4 Dof Modelmentioning

confidence: 99%

Resonant tori and instabilities in Hamiltonian systems

Litvak-Hinenzon

Rom‐Kedar

2002

Nonlinearity

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“…Geometry, transport, and chaos in Hamiltonian systems is a rich area of dynamical systems in which many tools have been developed. [6][7][8][9][10][11][12][13][14][15][16][17][18] It should be noted that Arnold diffusion, or global instability, [19][20][21] can take place in our system, since it possesses enough degrees of freedom. The effect of the presence of partial and complete barriers in the phase space and their effect on transport has many applications.…”

Section: Introductionmentioning

confidence: 99%

Geometry and transport in a model of two coupled quadratic nonlinear waveguides

Stirling

Bang

Christiansen

et al. 2008

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper applies geometric methods developed to understand chaos and transport in Hamiltonian systems to the study of power distribution in nonlinear waveguide arrays. The specific case of two linearly coupled ͑2͒ waveguides is modeled and analyzed in terms of transport and geometry in the phase space. This gives us a transport problem in the phase space resulting from the coupling of the two Hamiltonian systems for each waveguide. In particular, the effect of the presence of partial and complete barriers in the phase space on the transfer of intensity between the waveguides is studied, given a specific input and range of material properties. We show how these barriers break down as the coupling between the waveguides is increased and what the role of resonances in the phase space has in this. We also show how an increase in the coupling can lead to chaos and global transport and what effect this has on the intensity. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2840461͔We first present the general physical model for an arbitrary number of quadratic nonlinear waveguides, coupled linearly to each other. We normalize the system so that the dynamical evolution variable, i.e., the distance along the waveguides, can be big and therefore considered as a time variable, as is conventional in transport and chaos studies. We then focus on the two-core coupler (two waveguides) and investigate how structures in the phase space can lead to information about the coupling required for complete transfer of intensity from the fundamental wave (FW) in one waveguide to the second harmonic (SH; at twice the frequency) in the other waveguide. In particular, we investigate the structures that act as partial or complete barriers in the phase space. We study how these structures are destroyed as the coupling between the two waveguides is increased from the uncoupled regime to a critical value and beyond. The present work also includes a study of the effect of resonances in the phase space, as well as their connection with the destruction of complete barriers to transport. The sensitivity of the resulting intensity distribution to the strength of the coupling between the FW in the two waveguides "… is also studied. In particular, we show how the nature of the structures (or barriers) in the phase space can be changed with small changes in , such that a very ordered output can change to an output which appears ordered over a long time, but then suddenly becomes disordered or more complicated, to finally reach a very disordered state appearing to be the product of spatial-temporal chaos. Such behavior is of much practical importance, as it shows that, for certain values of , a small change in can produce very different outputs. This also shows that such sensitivity is inherent in the system and is not a product of noise or experimental error.

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“…It is seen that holding α 3 fixed in this limit changes the character of the energy surfaces from being bounded to being unbounded in I 1 , I 2 . We will not delve into the analysis of all the different limits which may be taken here; some of these limits are studied in detail in previous works (see [45,46,35,36,37,38]). In particular, note that the appearance of flat parabolic resonant tori in such a situation gives rise to strong instabilities (see [38] and [37]).…”

Section: Energy Surfaces In the Energy-momentum Space (B)mentioning

confidence: 99%

“…Other cases change some of the inequalities below, leading to a different EMBD and may be similarly analyzed; see [36,35,37,38], where we considered mainly unbounded models. Here, (7.5) and (7.6) define paraboloids, and their intersections with the plane of constant energy define ellipses (see Figures 10, 11, 12, 13, 14, 15, 16).…”

Section: Energy Surfaces In the Energy-momentum Space (B)mentioning

confidence: 99%

On Energy Surfaces and the Resonance Web

Litvak-Hinenzon¹,

Rom‐Kedar²

2004

SIAM J. Appl. Dyn. Syst.

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Abstract.A framework for understanding the global structure of near-integrable n DOF Hamiltonian systems is proposed. To this aim two tools are developed-the energy-momentum bifurcation diagrams and the branched surfaces. Their use is demonstrated on a few near-integrable 3 DOF systems. For these systems possible sources of instabilities are identified in the diagrams, and the corresponding energy surfaces are presented in the frequency space and by the branched surfaces. The main results of this formulation are theorems which describe the connection between changes in the topology of the energy surfaces and the existence of resonant lower dimensional tori. Key words. near-integrable Hamiltonians, parabolic resonances AMS subject classifications. 70H08, 37J20DOI. 10.1137/0306001061. Introduction. The study of the structure of energy surfaces of integrable systems and the study of resonances and instabilities in near-integrable systems developed into vast disparate research fields. The relation between the two received very little attention. Indeed, near regular level sets of the integrable Hamiltonian, the standard Arnold-resonance web structure appears, and the relation between the two fields reduces to the study of Arnold conjecture regarding instabilities in phase space. Here, we demonstrate that near singular level sets of the integrable Hamiltonian much information regarding possible instabilities of the near-integrable case may be deduced from the structure of the energy surface and its relations with resonance surfaces. We suggest that by adding some information to the traditional energy-momentum diagrams [7], which we name energy-momentum bifurcation diagrams (EMBD), one achieves a global qualitative understanding of the near-integrable dynamics. We relate the geometric properties of the surfaces corresponding to lower dimensional tori in this diagram to both bifurcations in the energy surface topology and the appearance of lower dimensional resonant tori.Recall that energy surfaces of generic integrable Hamiltonian systems are foliated almost everywhere by n-tori, 1 which may be expressed locally as a product of n circles on which the dynamics reduces to simple rotations (the action-angle coordinates). A given compact regular level set (the set of phase space points with given values of the constants of motion) may be composed of several such tori. The energy surface is composed of all level sets with the same

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Parabolic resonances in near integrable Hamiltonian systems

Cited by 4 publications

References 0 publications

Resonant tori and instabilities in Hamiltonian systems

Resonant tori and instabilities in Hamiltonian systems

Geometry and transport in a model of two coupled quadratic nonlinear waveguides

On Energy Surfaces and the Resonance Web

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