Consider the nonlinear system x˙ = v v˙ = −x − ϵx3 with a parameter ϵ. (a) Find a conserved quantity for this system. (b) Find and classify the fixed points of this system for ϵ = −1 and for ϵ = 1. Make sure to justify your classification. (c) Sketch phase portraits for this system for ϵ = −1 and for ϵ = 1. Please explain your process for creating your phase portraits (i.e. the steps that you take to make the picture).

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