Linear Versus Nonlinear Stability in Hamiltonian Systems

2017 ◽  
pp. 121-127 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Hamiltonian Systems ◽  
Nonlinear Stability

scholarly journals On the Effects of Circulation around a Circle on the Stability of a Thomson Vortex N-gon

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1033
Author(s):  
Leonid Kurakin ◽  
Irina Ostrovskaya
Keyword(s):  
Circular Cylinder ◽  
Hamiltonian Systems ◽  
Nonlinear Stability ◽  
Vortex Dynamics ◽  
Stability Problem ◽  
Orbital Stability ◽  
Stability And Instability ◽  
The Stability ◽  
Parameter Values ◽  
Linear Setting

The stability problem of the stationary rotation of N identical point vortices is considered. The vortices are located on a circle of radius R 0 at the vertices of a regular N-gon outside a circle of radius R. The circulation Γ around the circle is arbitrary. The problem has three parameters N, q, Γ , where q = R 2 / R 0 2 . This old problem of vortex dynamics is posed by Havelock (1931) and is a generalization of the Kelvin problem (1878) on the stability of a regular vortex polygon (Thomson N-gon) on the plane. In the case of Γ = 0 , the problem has already been solved: in the linear setting by Havelock, and in the nonlinear setting in the series of our papers. The contribution of this work to the solution of the problem consists in the analysis of the case of non-zero circulation Γ ≠ 0 . The linearization matrix and the quadratic part of the Hamiltonian are studied for all possible parameter values. Conditions for orbital stability and instability in the nonlinear setting are found. The parameter areas are specified where linear stability occurs and nonlinear analysis is required. The nonlinear stability theory of equilibria of Hamiltonian systems in resonant cases is applied. Two resonances that lead to instability in the nonlinear setting are found and investigated, although stability occurs in the linear approximation. All the results obtained are consistent with those known for Γ = 0 . This research is a necessary step in solving similar problems for the case of a moving circular cylinder, a model of vortices inside an annulus, and others.

Interaction of Lower and Higher Order Hamiltonian Resonances

2018 ◽  
Vol 28 (08) ◽  
pp. 1850097 ◽  
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Nonlinear Stability ◽  
Normal Forms ◽  
Double Resonance ◽  
Higher Order ◽  
Order Resonance ◽  
Resonance Zone ◽  
Stability Results ◽  
Low Order

The tools of normal forms and recurrence are used to analyze the interaction of low and higher order resonances in Hamiltonian systems. The resonance zones where the short-periodic solutions of the low order resonances exist are characterized by small variations of the corresponding actions that match the variations of the higher order resonance; this yields cases of embedded double resonance. The resulting interaction produces periodic solutions that in some cases destabilize a resonance zone. Applications are given to the three dof [Formula: see text] resonance and to periodic FPU-chains producing unexpected nonlinear stability results and quasi-trapping phenomena.

Deterministic Mechanism of Irreversibility

2018 ◽  
Vol 14 (3) ◽  
pp. 5708-5733 ◽  
Author(s):  
Vyacheslav Michailovich Somsikov
Keyword(s):  
Internal Energy ◽  
Hamiltonian Systems ◽  
Classical Mechanics ◽  
Motion Equation ◽  
Holonomic Constraints ◽  
Dissipative Processes ◽  
Motion Energy ◽  
Analytical Review ◽  
History Of ◽  
Material Points

The analytical review of the papers devoted to the deterministic mechanism of irreversibility (DMI) is presented. The history of solving of the irreversibility problem is briefly described. It is shown, how the DMI was found basing on the motion equation for a structured body. The structured body was given by a set of potentially interacting material points. The taking into account of the body’s structure led to the possibility of describing dissipative processes. This possibility caused by the transformation of the body’s motion energy into internal energy. It is shown, that the condition of holonomic constraints, which used for obtaining of the canonical formalisms of classical mechanics, is excluding the DMI in Hamiltonian systems. The concepts of D-entropy and evolutionary non-linearity are discussed. The connection between thermodynamics and the laws of classical mechanics is shown. Extended forms of the Lagrange, Hamilton, Liouville, and Schrödinger equations, which describe dissipative processes, are presented.

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