By Sinha, D. K.
An try and familiarize the reader with nonlinear structures, specifically qualitative features in quite a few structures amenable to mathematization.
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Additional info for Basics of nonlinearities in mathematical sciences
There is no hard and fast rule to find Liapunovs function. If the right hand side of the first system does not contain it, we should look for Liapunov function that does not depend on explicitly either. Indeed one has to go for trial and error method. Let some hunch be acquired through such exercises.
What is the 2 dt nature of the critical point in regard to its stability or otherwise? Where is it? Discuss the behaviour of the critical point of the system dy dx = −x , = −2 y dt dt Are these same as those of the system dx = 4y ; dt dy = −x dt ? Do they differ? If so, how? Linear Systems 47 11. Solve for x and y, in general terms, for the system dy dx = −x + 2 y =x, dt dt What is the critical point of the system? Study the behaviour of the system around the critical point. Draw a sketch of the paths for all possible cases.
4. How to go about with the findings of Liapunovs function? There is no hard and fast rule to find Liapunovs function. If the right hand side of the first system does not contain it, we should look for Liapunov function that does not depend on explicitly either. Indeed one has to go for trial and error method. Let some hunch be acquired through such exercises.