On the Resonance 1 – 1 for Hamiltonian Systems near an Equilibrium Point

1976 ◽  
pp. 345-345
Author(s):  
J. Roels
Keyword(s):  
Equilibrium Point ◽  
Hamiltonian Systems

scholarly journals The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems

1988 ◽  
Vol 8 (8) ◽  
pp. 87-97 ◽  
Keyword(s):  
Hamiltonian System ◽  
Equilibrium Point ◽  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Index Theory ◽  
Conley Index ◽  
At Resonance ◽  
Equivariant Version

AbstractAn equivariant version of Conley's homotopy index theory for flows is described and used to find periodic solutions of a Hamiltonian system locally near an equilibrium point which is at resonance.

Birkhoff normalization and superintegrability of Hamiltonian systems

2009 ◽  
Vol 29 (6) ◽  
pp. 1853-1880 ◽  
Author(s):  
HIDEKAZU ITO
Keyword(s):  
Hamiltonian System ◽  
Normal Form ◽  
Power Series ◽  
Holomorphic Function ◽  
Equilibrium Point ◽  
Hamiltonian Systems ◽  
Degree Of Freedom ◽  
Birkhoff Normal Form ◽  
Convergent Power Series ◽  
Liouville Integrable

AbstractWe study Birkhoff normalization in connection with superintegrability of ann-degree-of-freedom Hamiltonian systemXHwith holomorphic HamiltonianH. Without assuming any Poisson commuting relation among integrals, we prove that, if the system XHhasn+qholomorphic integrals near an equilibrium point of resonance degreeq≥0, there exists a holomorphic Birkhoff transformation φ such thatH∘φ becomes a holomorphic function ofn−qvariables and thatXH∘φcan be solved explicitly. Furthermore, the Birkhoff normal formH∘φ is determined uniquely, independently of the choice of φ, as convergent power series. We also show that the systemXHis superintegrable in the sense of Mischenko–Fomenko as well as Liouville-integrable near the equilibrium point.

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.

Dynamics of a prey predator disease model with Holling type functional response and stochastic perturbation

2020 ◽  
pp. 471-488
Author(s):  
M. N. Srinivas ◽  
G. Basava Kumar ◽  
V. Madhusudanan
Keyword(s):  
Equilibrium Point ◽  
Disease Model ◽  
Stochastic Perturbation ◽  
Positive Equilibrium ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Unique Positive Solution ◽  
Stochastic Nature ◽  
Research Article ◽  
Positive Equilibrium Point

The present research article constitutes Holling type II and IV diseased prey predator ecosystem and classified into two categories namely susceptible and infected predators.We show that the system has a unique positive solution. The deterministic and stochastic nature of the dynamics of the system is investigated. We check the existence of all possible steady states with local stability. By using Routh-Hurwitz criterion we showed that the positive equilibrium point $E_{7}$ is locally asymptotically stable if $x^{*} > \sqrt{m_{1}}$ .Moreover condition of the global stability of positive equilibrium point $E_{7}$ are also entrenched with help of Lyupunov theorem. Some Numerical simulations are carried out to illustrate our analytical findings.

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