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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 5

Question

Math 6410 • ODE Fall 1994 • Question 5

Let \(r^2=x^2+y^2\), \(u=r^4-3r^2+1\) and consider the planar system \(x'=-y+xu\), \(y'=x+yu\). Prove by means of the Poincaré–Bendixson theorem that there are two periodic orbits, one in \(0<r<1\) and another in \(1<r<3\).

Raw LaTeX

Let \(r^2=x^2+y^2\), \(u=r^4-3r^2+1\) and consider the planar
system \(x'=-y+xu\), \(y'=x+yu\). Prove by means of the Poincar\'e--Bendixson theorem
that there are two periodic orbits, one in \(0<r<1\) and another in \(1<r<3\).

Topic Tags

Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem Sobolev Spaces and Energy Estimates