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Describe how their relationship evolves with time. 3. Consider the following system describing the motion of a particle in a potential field ẋ = y y˙ = x2 − 1 a. Find all fixed points and classify them according to their linear stability. Compute eigenvectors for any saddles. b. Find V(x) such that the quantity E(x, y) = 2 y V (x) is conserved along 1 2 + trajectories. c. Sketch the phase portrait. d. Are there any periodic orbits? If yes, for what values of E? e. Is there a homoclinic orbit? If yes, for what value of E? 4. Show that the system ẋ = y − x3 y˙ = − x − y3 Has no periodic orbits. Justify your argument. (Hint: Use a Liapunov Function L(x, y) = ax2 + by2 with suitable a, b

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Differential Equations Questions: Motion, Linear Stability, System And Value
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