Does the unforced quartic oscillator behave chaotically?

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    Chaos Oscillator
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Discussion Overview

The discussion revolves around the behavior of the unforced quartic oscillator, particularly whether it exhibits chaotic dynamics. Participants explore the conditions under which chaos may arise, referencing the role of driving forces and stability of equilibria.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant suggests that the quartic oscillator is chaotic only when subjected to a driving force, referencing an external source.
  • Another participant notes that there are various mathematical definitions of chaos and questions which one is being applied in this context.
  • A different participant asserts that an autonomous Hamiltonian system with one degree of freedom is not chaotic in any reasonable sense.
  • Another participant clarifies that in Hamiltonian formalism, a system is defined to have degrees of freedom corresponding to the number of generalized coordinates and momenta.

Areas of Agreement / Disagreement

Participants express differing views on the chaotic nature of the unforced quartic oscillator, with no consensus reached on the definitions of chaos or the implications of Hamiltonian dynamics.

Contextual Notes

There are unresolved questions regarding the definitions of chaos and the specific conditions under which the quartic oscillator may exhibit chaotic behavior.

L_landau
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I thought that quartic oscillator is chaotic, but here
http://www.scholarpedia.org/article/Duffing_oscillator
it seems this is only when there is driving force.

It also says that for unforced quartic oscillator we can find equilibria and then determine if equilibria are stable or unstable. Is unstable equilibria chaotic?

Thanks
 
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There are a lot of different mathematical definitions of chaos. Which of them do you use?
Nevertheless, an autonomous Hamiltonian system with one degree of freedom is not chaotic in any reasonable sense
 
In Hamiltonian formalism there would be two degrees of freedom I thought?
 
In Hamiltonian formalism a system with Hamiltonian ##H=H(q_1,\ldots,q_m,p_1,\ldots,p_m)## by definition has ##m## degrees of freedom
 

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