{ x ˙ = − y − y 3 y ˙ = x where x , y ∈ R . Show that the critical point for the linear system is a center. Prove that the type of the critical point is the same for the nonlinear system.

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{                                                      x              ˙                                =          −          y          −                      y            3                                                                          y              ˙                                =          x                        where   x  ,  y  ∈      R  . Show that the critical point for the linear system is a center. Prove that the type of the critical point is the same for the nonlinear system.

Emely English · Accepted Answer

The differential equation is the same as       y          2        +  (  y  +      y    3    )  =  0, which describes free vibration of an oscillator with a nonlinear spring. The system is conserve in energy. So its orbit is a closed curve

Jaylyn Gibson · Accepted Answer

V  (  x  ,  y  )  =  1      /    2      x    2    +  1      /    2      y    2    +  1      /    4      y    4    ,which is positive definite and   V  (  x  =  0  ,  y  =  0  )  =  0The derivative is given by            V      ˙        =  x            x      ˙        +  y            y      ˙        +      y    3              y      ˙        =  x  [  −  y  −      y    3    ]  +  y  x  +      y    3    [  x  ]  ≡  0.This condition implies that the origin is a center.

We have {dot x=−y−y^3; dot y=x. Where x,y in R. Show that the critical point for the linear system is a center. Prove that the type of the critical point is the same for the nonlinear system.

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