Linear stability of symplectic maps

Howard, James E.; MacKay, Robert S.

doi:10.1063/1.527544

Cited by 74 publications

(59 citation statements)

References 13 publications

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“…These results are a consequence of general theorems of Moser and Krein respectively [7] and of additional explicit calculations [8]. We apply them on the effect of perturbation of the transversal multipliers in proposition 4.5 to obtain Proposition 5.2.…”

Section: Behavior Of Multipliers Under Perturbationmentioning

confidence: 99%

“…In a situation in which multipliers of periodic orbits of Hamiltonian flows (nearly) coincide, Krein signature determines largely if a perturbation leads to instability or not [7]. This theory can be extended to Hamiltonian systems with friction [8].…”

Section: Behavior Of Multipliers Under Perturbationmentioning

confidence: 99%

“…The effect of coupling between dominant and transversal oscillators can be described with the aid of the formalism of Krein signatures for Hamiltonian systems [6,7] which can be extended to include friction [8]. This is done in section 5.…”

Section: Introductionmentioning

confidence: 99%

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Period doubling solitons: yes or no?

Valkering

Zeegers

1991

Physica D: Nonlinear Phenomena

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We investigate the possibility of period doubling transition to temporal chaos of a coherent structure in a linile dimensional ttamiltonian system with moderately small friction. A solitary' wave in a periodically driven chain of particles provides a typical example. To this end the equations of motion are transformed to those of one driven dominant oscillator with one degree of freedom, weakly coupled to a set of oscillators representing the other degrees of freedom. The equations for the dominant oscillator alone contirm the possibility of such a transition to chaos tk~r well chosen driving. However, taking into account coupling to the other degrees of freedom, one must conclude that the supposed infinite period doubling sequence for small or no dampling snt~'ives only in a heavily damaged state. In practice a reproducable completed sequence cannot be expected unless the damping is strong enough.

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Section: Behavior Of Multipliers Under Perturbationmentioning

confidence: 99%

Section: Behavior Of Multipliers Under Perturbationmentioning

confidence: 99%

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Period doubling solitons: yes or no?

Valkering

Zeegers

1991

Physica D: Nonlinear Phenomena

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“…The Z2-symmetries in our pendulum system can be expressed as symplectic matrices with respect to the same matrix J [17]. The product of two symplectic matrices is again a symplectic matrix, and its eigenvalues appear either as complex pairs on the unit circle (2,2, with 1)~1 = 1), as real pairs (2,2 J), or as quadruplets (2,2,).-I,.~-t).…”

Section: Lrjl=jmentioning

confidence: 99%

“…The symplectic character of Hamiltonian systems guarantees [17,18] that M is a symplectic map. 4 This means that the Jacobian matrix of M is symplectic with respect to the standard symplectic matrix J, i.e.,…”

Section: Period Doubling Bij~trcationsmentioning

confidence: 99%

Mode competition in a system of two parametrically driven pendulums; the role of symmetry

Banning

Weele

Kettenis

1997

Physica A: Statistical Mechanics and its Applications

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This paper is the final part in a series of four on the dynamics of two coupled, parametrically driven pendulums. In the previous three parts (Banning and van der Weele, Mode competition in a system of two parametrically driven pendulums; the Hamiltonian case, Physica A 220 (1995) 485-533; Banning et al., Mode competition in a system of two parametrically driven pendulums; the dissipative case, Physica A 245 (1997) l 1-48; Banning et al., Mode competition in a system of two parametrically driven pendulums with nonlinear coupling, Physica A 245 (1997) 49-98) we have given a detailed survey of the different oscillations in the system, with particular emphasis on mode interaction. In the present paper we use group theory to highlight the role of symmetry. It is shown how certain symmetries can obstruct period doubling and Hopf bifurcations; the associated routes to chaos cannot proceed until these symmetries have been broken. The symmetry approach also reveals the general mechanism of mode interaction and enables a useful comparison with other systems.

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Signatures for ‐hermitians and ‐unitaries on Krein spaces with Real structures

Schulz-Baldes

Villegas-Blas²

2016

Mathematische Nachrichten

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For J‐hermitian operators on a Krein space (K,J) satisfying an adequate Fredholm property, a global Krein signature is shown to be a homotopy invariant. It is argued that this global signature is a generalization of the Noether index. When the Krein space has a supplementary Real structure, the sets of J‐hermitian Fredholm operators with Real symmetry can be retracted to certain of the classifying spaces of Atiyah and Singer. Secondary Z2‐invariants are introduced to label their connected components. Related invariants are also analyzed for J‐unitary operators.

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Linear stability of symplectic maps

Cited by 74 publications

References 13 publications

Period doubling solitons: yes or no?

Period doubling solitons: yes or no?

Mode competition in a system of two parametrically driven pendulums; the role of symmetry

Signatures for ‐hermitians and ‐unitaries on Krein spaces with Real structures

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