# A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

@article{Hu2020ADB,
title={A dihedral Bott-type iteration formula and stability of symmetric periodic orbits},
author={Xijun Hu and Alessandro Portaluri and Ran Yang},
journal={Calculus of Variations and Partial Differential Equations},
year={2020},
volume={59},
pages={1-40}
}
• Published 25 May 2017
• Mathematics
• Calculus of Variations and Partial Differential Equations
Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant Hamiltonian systems. We apply the abstract theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Finally we provide a hyperbolicity criterion for reversible Lagrangian systems.
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## References

SHOWING 1-10 OF 55 REFERENCES
Bott formula of the Maslov-type index theory
In this paper the integer valued !-index theory parameterized by all ! on the unit circle for paths in the symplectic group Sp(2n) is established. Based on this index theory, the Bott formula of the
The iteration formulae of the Maslov-type index theory in weak symplectic Hilbert space
• Mathematics
• 2016
The authors prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then a direct proof of the iteration
Index and Stability of Symmetric Periodic Orbits in Hamiltonian Systems with Application to Figure-Eight Orbit
• Mathematics
• 2009
In this paper, using the Maslov index theory in symplectic geometry, we build up some stability criteria for symmetric periodic orbits in a Hamiltonian system, which is motivated by the recent
Seifert Conjecture in the Even Convex Case
• Mathematics
• 2013
In this paper, we prove that there exist at least n geometrically distinct brake orbits on every C2 compact convex symmetric hypersurface Σ in ℝ2n satisfying the reversible condition NΣ = Σ with N =
Spectral flow, crossing forms and homoclinics of Hamiltonian systems
We prove a spectral flow formula for one‐parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of
The homology of path spaces and Floer homology with conormal boundary conditions
• Mathematics
• 2008
Abstract.We define the Floer complex for Hamiltonian orbits on the cotangent bundle of a compact manifold which satisfy non-local conormal boundary conditions. We prove that the homology of this
Maslov-Type Index Theory for Symplectic Paths and Spectral Flow (II)
• Mathematics
• 1980
Based on the spectral flow and the stratification structures of the symplectic group Sp(2n, C), the Maslov-type index theory and its generalization, the ω-index theory parameterized by all ω on the
On a Generalized Sturm Theorem
Abstract Sturm oscillation theorem for second order differential equations was generalized to systems and higher order equations with positive leading coefficient by several authors. What we propose
Index theory for heteroclinic orbits of Hamiltonian systems
• Mathematics
• 2017
Index theory revealed its outstanding role in the study of periodic orbits of Hamiltonian systems and the dynamical consequences of this theory are enormous. Although the index theory in the periodic
On the Maslov index of symplectic paths that are not transversal to the Maslov cycle. Semi-Riemannian index theorems in the degenerate case
• Mathematics
• 2003
We use the notion of generalized signatures at a singularity of a smooth curve of symmetric bilinear forms to determine a formula for the computation of the Maslov index in the case of a