Question

Consider the system * =y j= +1. Find the fixed points and the linearisation of the system at each. Identify the type and sketch a local phase portrait (i.e. a sketch of the orbits just around the fixed points) at each fixed point. Show that the system has a time reversal symmetry. Draw a sketch showing isoclines and the directions of orbits in all parts of phase space. Use this information, together with the symmetry, to show there exists a homoclinic orbits. Sketch the global phase portrait. (HINT: In this case the stability properties of the fixed points and the time reversal symmetry should be sufficient to draw the direction diagram without explicitly evaluating signs of y and i on isoclines. However, it is also correct to proceed this latter way).

Answer 1

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phase pontrait YJ Systen af (-1,0 is Now the linearization 1 l af of l 11&l 1-X 11 20 Now the coeffierent matry, (b) the characteristic equation is 1-2 1-1-x)" = x+120 to the the eigen rake are 2x z Ii zdeiß purely imaginary. so the nature of the point in 220, B2+1 do so stable centre phase portrait