Iteration formulae for brake orbit and index inequalities for real pseudoholomorphic curves

Zhou, Beijia

doi:10.1007/s11784-021-00928-3

Cited by 3 publications

(1 citation statement)

References 24 publications

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“…In particular, while the value of b is not an invariant, its sign is an invariant. Following [10] we refer to sign(b) as the B-sign of the reduced monodromy, see also [25]. In the case elliptic case, by [10, Appendix B], the B-sign gives the same information as the Krein type of the eigenvalues of the reduced monodromy (as introduced in [15,16,17,18,19]).…”

Section: Introductionmentioning

confidence: 99%

On doubly symmetric periodic orbits

Frauenfelder

Moreno

2023

Celest Mech Dyn Astron

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In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e., those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) All covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory (Eliashberg et al. Geom Funct Anal Special Volume Part II:560–673, 2000); (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in Frauenfelder et al. in Symplectic methods in the numerical search of orbits in real-life planetary systems. Preprint arXiv:2206.00627). The above results follow from: (5) A symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in Frauenfelder and Moreno 2021) differ.

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Section: Introductionmentioning

confidence: 99%

On doubly symmetric periodic orbits

Frauenfelder

Moreno

2023

Celest Mech Dyn Astron

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The Conley–Zehnder indices of the spatial Hill three-body problem

Aydin

2023

Celest Mech Dyn Astron

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We explore the interaction between the Conley–Zehnder index and bifurcation points of symmetric planar as well as spatial periodic orbits in the spatial Hill three-body problem. We start with the fundamental families of planar periodic orbits which are those of direct (family g) and retrograde periodic orbits (family f). Since the spatial system is invariant under a symplectic involution, whose fixed point set corresponds to the planar problem, planar orbits have planar and spatial Floquet multipliers, and planar and spatial Conley–Zehnder indices. When the Floquet multipliers move through a root of unity, new families of periodic orbits bifurcate and the index jumps. For very low energies, the families g and f arise dynamically from the rotating Kepler problem, and in a recent work (Aydin From Babylonian lunar observations to Floquet multipliers and Conley-Zehnder Indices) we determined analytically their indices. By their numerical continuations for higher energies, we determine the index of various families of planar and spatial periodic orbits bifurcating from g and f. Since these families can bifurcate again and meet each other, this procedure can get complicated. This index leads to a grading on local Floer homology. Since the local Floer homology and its Euler characteristic stay invariant under bifurcation, the index provides important information about the interconnectedness of such families, which we illustrate in form of bifurcation graphs. Since the solutions of Hill’s system may serve as orbits for space mission design or astronomical observations, our results promote the interaction between Symplectic Geometry and practical problems.

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On doubly symmetric periodic orbits

Frauenfelder

Moreno²

2023

Preprint

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In this article, for Hamiltonian systems with two degrees of freedom, we study doubly symmetric periodic orbits, i.e. those which are symmetric with respect to two (distinct) commuting antisymplectic involutions. These are ubiquitous in several problems of interest in mechanics. We show that, in dimension four, doubly symmetric periodic orbits cannot be negative hyperbolic. This has a number of consequences: (1) all covers of doubly symmetric orbits are good, in the sense of Symplectic Field Theory [6]; (2) a non-degenerate doubly symmetric orbit is stable if and only if its CZ-index is odd; (3) a doubly symmetric orbit does not undergo period doubling bifurcation; and (4) there is always a stable orbit in any collection of doubly symmetric periodic orbits with negative SFT-Euler characteristic (as coined in [11]). The above results follow from: (5) a symmetric orbit is negative hyperbolic if and only its two B-signs (introduced in [10]) differ.

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Iteration formulae for brake orbit and index inequalities for real pseudoholomorphic curves

Cited by 3 publications

References 24 publications

On doubly symmetric periodic orbits

On doubly symmetric periodic orbits

The Conley–Zehnder indices of the spatial Hill three-body problem

On doubly symmetric periodic orbits

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