Conservative system ODE

Consider the following system:
̇x = y
̇y = −x + x3 + ay
a) (1 pts) Find the fixed points of the system.
b) (1.25 pts) For which values of a is the system not conservative? Why?
c) (1 pts) Show that for the other values of a, E(x, y) = (-1/2)y^2-(1/2)x^2+(1/4)x^4 is a conserved quantity.
d) (1.25 pts) Study the stability of the fixed points.
(Hint: Make sure that you study the non-linear stability of each point.)
e) (1 pts) Is the system reversible? Why?

Ordinary Differential Equations
Eva May Eva May
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  • Savionf Savionf
    +2

    The offered bounty is low for a five-part question.

    • Eva May Eva May
      0

      I increased it

    • Savionf Savionf
      +2

      Still low. Answering this question will take over an hour. Think about how much the time an individual skilled to answer this question is worth. That would give you an idea for a fair bounty.

  • Mathe Mathe
    0

    What do they mean by non-linear stability? Would checking the eigenvalues of the linearized system be enough?

    • Eva May Eva May
      0

      Im not entirely sure but I think that you are correct about checking the eigenvalues

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Mathe Mathe
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  • Eva May Eva May
    0

    Could you solve e) as well

    • Mathe Mathe
      0

      Oh sorry I missed that. One second!

    • Mathe Mathe
      0

      The second file (same name) should have the complete solution!

The answer is accepted.
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