arXiv subjects

Duanzhi Zhang

Publications and source records attributed to Duanzhi Zhang.

8 recordsLinked to original sources

arXiv2023

Infinitely many Brake orbits of Tonelli Hamiltonian systems on the cotangent bundle

We prove that on the twisted cotangent bundle of a closed manifold with an exact magnetic form, a Hamiltonian system of a time-dependent Tonelli Hamiltonian function possesses infinitely many brake orbits. More precisely, by applying Legendre transform we show that there are infinitely many symmetric orbits of the dual Euler-Lagrange system on the configuration space. This result contains an assertion for the existence of infinitely many symmetric orbits of Tonelli Euler-Lagrange systems given by G. Lu at the end of [Lu09a, Remark 6.1]. In this paper, we will present a complete proof of this assertion.

math.DS↗

arXiv2019

Elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\bf R}^{2n}$

Let $Σ$ be a compact convex hypersurface in ${\bf R}^{2n}$ which is P-cyclic symmetric, i.e., $x\in Σ$ implies $Px\inΣ$ with P being a $2n\times2n$ symplectic orthogonal matrix and $P^k=I_{2n}$, where $n, k\geq2$, $ker(P-I_{2n})=0$. In this paper, we first generalize Ekeland index theory for periodic solutions of convex Hamiltonian system to a index theory with P boundary value condition and study its relationship with Maslov P-index theory, then we use index theory to prove the existence of elliptic and non-hyperbolic closed characteristics on compact convex P-cyclic symmetric hypersurfaces in ${\bf R}^{2n}$ for a broad class of symplectic orthogonal matrix P.

math.DS↗

arXiv2015

Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-κ},I_κ,-I_{n-κ},I_κ)$ for some integer $κ\in[0, n-1)$, and $Σ\subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in Σ$ implies $Px\inΣ$. We prove that if $Σ$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $Σ$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4κ$ Floquet multipliers on the unit circle of the complex plane.

math.DS↗

arXiv2014

On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in ${\bf R}^{2n}$

In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-κ},I_κ,-I_{n-κ},I_κ)$ for some integer $κ\in[0, n)$, and $Σ\subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in Σ$ implies $Px\inΣ$. We prove that if $Σ$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{2}$, then there exist at least $n-κ$ geometrically distinct P-symmetric closed characteristics on $Σ$, as a consequence, $Σ$ carry at least $n$ geometrically distinct P-invariant closed characteristics.

math.DS↗

arXiv2013

Seifert conjecture in the even convex case

In this paper, we prove that there exist at least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a consequence, we show that if the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer $n$.

math.DS↗

arXiv2011

Multiple brake orbits on compact convex symmetric reversible hypersurfaces in $\R^{2n}$

In this paper, we prove that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $\Sg$ in $\R^{2n}$ for $n\ge 2$ satisfying the reversible condition $N\Sg=\Sg$ with $N=\diag (-I_n,I_n)$. As a consequence, we show that there exist at least $[\frac{n+1}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\R^{n}$ with $n\ge 2$ which gives a positive answer to the Seifert conjecture of 1948 in the symmetric case for $n=3$. As an application, for $n=4$ and 5, we prove that if there are exactly $n$ geometrically distinct closed characteristics on $\Sg$, then all of them are symmetric brake orbits after suitable time translation.

math.DS↗

arXiv2011

Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0=\{0\}\times \R^n\subset \R^{2n}$ and $L_1=\R^n\times \{0\} \subset \R^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\R^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H"_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$

math.DS↗

arXiv2011

Iteration theory of $L$-index and Multiplicity of brake orbits

In this paper, we first establish the Bott-type iteration formulas and some abstract precise iteration formulas of the Maslov-type index theory associated with a Lagrangian subspace for symplectic paths. As an application, we prove that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface $Σ$ in $\mathbb{R}^{2n}$ satisfying the reversible condition $NΣ=Σ$, furthermore, if all brake orbits on this hypersurface are nondegenerate, then there are at least $n$ geometrically distinct brake orbits on it. As a consequence, we show that there exist at least $[\frac{n}{2}]+1$ geometrically distinct brake orbits in every bounded convex symmetric domain in $\mathbb{R}^{n}$, furthermore, if all brake orbits in this domain are nondegenerate, then there are at least $n$ geometrically distinct brake orbits in it. In the symmetric case, we give a positive answer to the Seifert conjecture of 1948 under a generic condition.

math.SG↗