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- Consider the system (x' = 2xy - 2y ly' = x - y² a) Check that the system is Hamiltonian. (Consequence: it will not have asymptotically stable equilibrium points or repulsors nor limit cycles.) b) Study the local behavior of the equilibrium points linearizing the system. c) Find the equation of the trajectories of the form H(x, y) = C. Find the tangent curves (separatrices) to the stable spaces E³ and unstable E" of the linearization about each equilibrium point (x,y) that is not a center using the level curves H(x, y) = H(x,y). Make a sketch of the trajectories. Check the result with the maxima (or other program)

- Consider the system (x' = 2xy - 2y ly' = x - y² a) Check that the system is Hamiltonian. (Consequence: it will not have asymptotically stable equilibrium points or repulsors nor limit cycles.) b) Study the local behavior of the equilibrium points linearizing the system. c) Find the equation of the trajectories of the form H(x, y) = C. Find the tangent curves (separatrices) to the stable spaces E³ and unstable E" of the linearization about each equilibrium point (x,y) that is not a center using the level curves H(x, y) = H(x,y). Make a sketch of the trajectories. Check the result with the maxima (or other program)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q3. Consider the system (x' = 2xy - 2y ly' = x - y² a) Check that the system is Hamiltonian. (Consequence: it will not have asymptotically stable equilibrium points or repulsors nor limit cycles.) b) Study the local behavior of the equilibrium points linearizing the system. c) Find the equation of the trajectories of the form H(x, y) = C. Find the tangent curves (separatrices) to the stable spaces E³ and unstable E" of the linearization about each equilibrium point (x,y) that is not a center using the level curves H(x, y) = H(x,y). Make a sketch of the trajectories. Check the result with the maxima (or other program)
Transcribed Image Text:Q3. Consider the system (x' = 2xy - 2y ly' = x - y² a) Check that the system is Hamiltonian. (Consequence: it will not have asymptotically stable equilibrium points or repulsors nor limit cycles.) b) Study the local behavior of the equilibrium points linearizing the system. c) Find the equation of the trajectories of the form H(x, y) = C. Find the tangent curves (separatrices) to the stable spaces E³ and unstable E" of the linearization about each equilibrium point (x,y) that is not a center using the level curves H(x, y) = H(x,y). Make a sketch of the trajectories. Check the result with the maxima (or other program)
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