On the index theories for second order Hamiltonian systems

An, Tianqing; Ye, Long

doi:10.1016/s0362-546x(97)00572-5

Cited by 18 publications

(24 citation statements)

References 6 publications

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“…Note that simultaneously published with the topological extension of results in [3] and [18] to the degenerate Hamiltonian systems in [9] of 1990, the Conley-Zehnder's index theory for non-degenerate symplectic paths in [3] was also extended to the degenerate Hamiltonian systems by an analytic method of C. Viterbo in [22]. Note also that when x is a solution of a second order Hamiltonian system, Theorem 1.4 is proved in [1] by a rather simple variational method. Note also that when x is a solution of a second order Hamiltonian system, Theorem 1.4 is proved in [1] by a rather simple variational method.…”

Section: Hyperbolic Extremals In the Calculus Of Variationmentioning

confidence: 96%

Indexing domains of instability for Hamiltonian systems

Ye¹,

An²

1998

NoDEA : Nonlinear Differential Equations and Applications

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Based upon the understanding of the global topologies of the singular subset, its complement, and the hyperbolic subset in the symplectic group, in this paper we study the domains of instability for hyperbolic Hamiltonian systems and define a characteristic index for such domains. This index is defined via the Maslov-type index theory for symplectic paths starting from the identity defined by C. Conley, E. Zehnder, and Y. Long, and the hyperbolic index of symplectic matrices. The old problem of the relation between the nondegenerate local minimality and the instability of hyperbolic extremal loops in the calculus of variation is also studied via this new index for the domains of instability. * Partially supported by the NNSF and MCSEC of China and the Qiu Shi Sci. Tech. Foundation. † Associate member of ICTP 462 Yiming Long and Tianquing An NoDEA Definition 1.1 For any matrix M ∈ Sp(2n), the hyperbolic index (M ) of M is defined to be the mod 2 number of the total multiplicity of negative eigenvalues of M which are strictly less than −1.We consider the following linear Hamiltonian systems with continuous periodic symmetric coefficients:ż = JB(t)z, (1.1)where B ∈ C(S 1 , L s (R 2n )), S 1 = R/Z, Z denotes the integer set, and · denotes the derivative d dt . Let H(2n) be the set of all such linear Hamiltonian systems. Denote by γ B the fundamental solution of (1.1) starting from the identity. Then γ B ∈ P(2n). The eigenvalues of γ B (1) are called the Floquet multipliers of the system (1.1). Definition 1.2 A matrixM ∈ Sp(2n) is hyperbolic, if all the eigenvalues of M are not on the unit circle U in the complex plane C except two of them which are 1. Denote by Sp h (2n) the subset of all hyperbolic matrices in Sp 2n). Denote by P h (2n) the subset of all hyperbolic paths in P(2n). A system (1.1) in H(2n) is hyperbolic, if its fundamental solution γ B ∈ P h (2n). Denote by H h (2n) the subset of all hyperbolic systems in H(2n). In this case we also write B ∈ H h (2n).Note that this concept of the hyperbolicity comes from the study of periodic solutions of autonomous Hamiltonian systems. As we shall show below, P h (2n) and H h (2n) contain countably infinitely many path connected components which are called the domains of instability of symplectic paths and linear Hamiltonian systems respectively. Definition 1.3 Two paths γ 0 and γ 1 ∈ P h (2n) belong to the same domain of instability, if γ 0 and γ 1 can be connected by a continuous one-parameter family of paths {γ s } 0≤s≤1 in P h (2n). In this case we write γ 0 ∼ h γ 1 . Two Hamiltonian systems of the form (1.1) with B i ∈ C(S 1 , L s (R 2n )) ∩ H h (2n) for i = 0, 1 belong to the same domain of instability, if B 0 and B 1 can be connected by a continuous one-parameter family of {B s } 0≤s≤1 in C(S 1 , L s (R 2n )) ∩ H h (2n). In this case we write B 0 ∼ h B 1 .Note that for γ 0 and γ 1 ∈ P h (2n), the facts that γ 0 ∼ γ 1 on [0, 1] along δ(·, 1) in the sense of Definition 4.2 below and that δ(·, 1) being a path in Sp h (2n) together imply γ 0 ∼ h γ 1 .A powerful t...

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Section: Hyperbolic Extremals In the Calculus Of Variationmentioning

confidence: 96%

Indexing domains of instability for Hamiltonian systems

Ye¹,

An²

1998

NoDEA : Nonlinear Differential Equations and Applications

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“…(3) The right equality of (1.4) holds for some m>1 if and only if there holds I 2p h N 1 (1,1) h r h K # 0 0 (#({)) for some non-negative integers p and r satisfying p+r n and some K # Sp(2(n& p&r)) with _(K)/U"R satisfying the following conditions:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…If m=2, there holds (&I 2s ) h N 1 (&1, 1) h t h H # 0 0 (K) for some non-negative integers s and t satisfying 0 s+t n& p&r, and some H # Sp(2(n& p&r&s&t)) satisfying _(H)/U"R and that all elements in (2) The inequality (1.3) in Theorem 1.1 was first proved in our other paper [13]. The necessary and sufficient conditions for the equalities in Theorem 1.1 are new.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Suppose the left equality of (3.1) holds for a given | # U + " [1]. When q=0 in (3.2), all equalities must hold in (3.4) and (3.5).…”

Section: Proofs Of the Iteration Inequalitiesmentioning

confidence: 99%

“…When q=0, since | can range over all of U + " [1], by Part 2 all the eigenvalues of #({) must be equal to 1 and (3.10) (3.12) hold. This implies & { (#)=2n.…”

Section: Proofs Of the Iteration Inequalitiesmentioning

confidence: 99%

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Iteration Inequalities of the Maslov-Type Index Theory with Applications

Liu

2000

Journal of Differential Equations

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In this paper, by using the |-index theory introduced by Y. Long in (1999, Pacific J. Math. 187, 113 149), in particular, the splitting numbers, Maslov-type mean index, and the homotopy component of symplectic matrix, we establish various inequalities of the Maslov-type index theory for iterations of symplectic paths starting from the identity. As an application, these results are used to study Rabinowitz' conjecture on the prescribed minimal period solution problem of nonlinear Hamiltonian systems. Academic Press

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The L-Index Theory

Liu

2019

Index Theory in Nonlinear Analysis

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On the index theories for second order Hamiltonian systems

Cited by 18 publications

References 6 publications

Indexing domains of instability for Hamiltonian systems

Indexing domains of instability for Hamiltonian systems

Iteration Inequalities of the Maslov-Type Index Theory with Applications

The L-Index Theory

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