Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuang Wang ◽  
Dingbian Qian
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Phase Plane Analysis ◽  
Cauchy Problems ◽  
Multiplicity Results ◽  
Birkhoff Theorem ◽  
Subharmonic Solutions ◽  
Large Solutions ◽  
Plane Analysis ◽  
Rotation Numbers

Abstract We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

scholarly journals Subharmonic solutions of bounded coupled Hamiltonian systems with sublinear growth

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fanfan Chen ◽  
Dingbian Qian ◽  
Xiying Sun ◽  
Yinyin Wu
Keyword(s):  
Hamiltonian Systems ◽  
Phase Plane ◽  
Phase Plane Analysis ◽  
Zero Eigenvalue ◽  
Subharmonic Solutions ◽  
Existence And Multiplicity ◽  
Plane Analysis ◽  
Higher Dimensional ◽  
Dimensional Version ◽  
Twist Theorem

<p style='text-indent:20px;'>We prove the existence and multiplicity of subharmonic solutions for bounded coupled Hamiltonian systems. The nonlinearities are assumed to satisfy Landesman-Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is based on phase plane analysis and a higher dimensional version of the Poincaré-Birkhoff twist theorem by Fonda and Ureña. The results obtained generalize the previous works for scalar second-order differential equations or relativistic equations to higher dimensional systems.</p>

Subharmonic Solutions of Planar Hamiltonian Systems: a Rotation Number Approach

2011 ◽  
Vol 11 (1) ◽  
Author(s):  
Alberto Boscaggin
Keyword(s):  
Hamiltonian System ◽  
Hamiltonian Systems ◽  
Rotation Number ◽  
Subharmonic Solutions ◽  
Planar Hamiltonian System ◽  
Nodal Properties

AbstractWe prove the existence of infinitely many subharmonic solutions, with prescribed nodal properties, for a planar Hamiltonian system Jz′ = Δ

scholarly journals Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

2017 ◽  
Vol 8 (1) ◽  
pp. 583-602 ◽  
Author(s):  
Alessandro Fonda ◽  
Rodica Toader
Keyword(s):  
Hamiltonian Systems ◽  
Zero Eigenvalue ◽  
Birkhoff Theorem ◽  
Subharmonic Solutions ◽  
Existence And Multiplicity ◽  
Volterra Systems

Abstract We prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations of N planar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

scholarly journals Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus

1988 ◽  
Vol 8 (4) ◽  
pp. 555-584 ◽  
Author(s):  
Raphaël Douady
Keyword(s):  
Hamiltonian Systems ◽  
Rotation Number ◽  
Positive Measure ◽  
Invariant Curves ◽  
Rotation Numbers ◽  
Twist Maps ◽  
Area Preserving Maps ◽  
Area Preserving

AbstractWe prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.

Whole-body phase plane analysis for standard maximum vertical jump assessment

2021 ◽  
Vol 90 ◽  
pp. 203-204
Author(s):  
C. Rodrigues ◽  
M. Correia ◽  
J. Abrantes ◽  
B. Rodrigues ◽  
J. Nadal
Keyword(s):  
Phase Plane ◽  
Vertical Jump ◽  
Whole Body ◽  
Phase Plane Analysis ◽  
Plane Analysis

Influence of Radial Rotation on Heat Transfer in a Rectangular Channel With Two Opposite Walls Roughened by Hemispherical Protrusions at High Rotation Numbers

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Shyy Woei Chang ◽  
Tong-Miin. Liou ◽  
Wei-Chun Chen
Keyword(s):  
Heat Transfer ◽  
Infrared Thermography ◽  
Rotation Number ◽  
Rectangular Channel ◽  
Flow Region ◽  
Transfer Data ◽  
Nusselt Numbers ◽  
Rotation Numbers ◽  
Heat Transfer Correlations ◽  
Rotating Channel

Detailed heat transfer distributions over two opposite leading and trailing walls roughened by hemispherical protrusions were measured from a rotating rectangular channel at rotation number up to 0.6 to examine the effects of Reynolds (Re), rotation (Ro), and buoyancy (Bu) numbers on local and area-averaged Nusselt numbers (Nu and Nu¯) using the infrared thermography. A set of selected heat transfer data illustrates the Coriolis and rotating buoyancy effects on the detailed Nu distributions and the area-averaged heat transfer performances of the rotating channel. The Nu¯ for the developed flow region on the leading and trailing walls are parametrically analyzed to devise the empirical heat transfer correlations that permit the evaluation of the interdependent and individual Re, Ro, and Bu effect on Nu¯.

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