Hamiltonian Systems: Showing Limit Cycles Impossible

In summary: A limit cycle is a closed trajectory x(t) that is both periodic and asymptotically stable, meaning that x(t) approaches the cycle as t -> inf. In summary, a periodic solution is a repeating trajectory and a limit cycle is a stable, closed repeating trajectory in a Hamiltonian system.
  • #1
Nusc
760
2
http://books.google.ca/books?id=Pd8...ook_result&ct=result&resnum=1&ved=0CAkQ6AEwAA


Let ~ denote vector.
For a fixed point.
(~x0 is asymptotically stable if there exists
a neighbourhood U of ~x0 such that if ~x(t) obeys Hamilton's equations and ~x(0) in U, then lim
t->inf
~x(t) = ~x0.)

can you give me a precise definition for periodic solutions (limit cycles) in this context?
 
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  • #2
Nusc said:
http://books.google.ca/books?id=Pd8...ook_result&ct=result&resnum=1&ved=0CAkQ6AEwAA


Let ~ denote vector.
For a fixed point.
(~x0 is asymptotically stable if there exists
a neighbourhood U of ~x0 such that if ~x(t) obeys Hamilton's equations and ~x(0) in U, then lim
t->inf
~x(t) = ~x0.)

can you give me a precise definition for periodic solutions (limit cycles) in this context?
A solution x is periodic iff there exists T >0 with x(t+T) = x(t) for all t.
 

What is a Hamiltonian System?

A Hamiltonian system is a mathematical model used to describe the behavior of a physical system over time. It involves the use of Hamiltonian equations, which are a set of differential equations that describe the evolution of a system's state variables.

What are Limit Cycles in Hamiltonian Systems?

Limit cycles are periodic orbits in a Hamiltonian system where the system's state variables return to their original values after a certain amount of time. They are characterized by their stability, and are often used to study the long-term behavior of a system.

Why is it Difficult to Show Limit Cycles are Impossible in Hamiltonian Systems?

It is difficult to show that limit cycles are impossible in Hamiltonian systems because of the complexity and non-linearity of these systems. Many Hamiltonian systems have multiple state variables and the interactions between them can be difficult to analyze mathematically.

What Methods are Used to Study Limit Cycles in Hamiltonian Systems?

There are several methods used to study limit cycles in Hamiltonian systems, including numerical simulations, bifurcation analysis, and perturbation techniques. Each method has its own advantages and limitations, and the choice of method depends on the specific characteristics of the system being studied.

Why are Limit Cycles Important in the Study of Hamiltonian Systems?

Limit cycles are important in the study of Hamiltonian systems because they can provide insights into the long-term behavior of a system. They can also help identify regions of stability and instability, and can be used to predict the occurrence of other types of behaviors, such as chaos.

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