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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 7

Question

Math 6410 • ODE Fall 1996 • Question 7

Let \(f\) have the special form \[ f(t,x)=Ax+h(t), \tag {2.10} \] where \(h:\mathbb {R}\to \mathbb {R}^N\) is a continuous function of period \(T\) and \(A\) is a constant \(N\times N\) matrix which is positive (negative) definite (i.e. \(Ax\cdot x\ge \mu \lvert x\rvert ^2\), resp. \(Ax\cdot x\le -\mu \lvert x\rvert ^2\), \(\mu >0\)). Use the result proved above to show that the equation \[ x'=Ax+h(t) \tag {2.11} \] has a unique \(T\)-periodic solution.

Raw LaTeX

Let \(f\) have the special form
\[
f(t,x)=Ax+h(t),
\tag{2.10}
\]
where \(h:\R\to\R^N\) is a continuous function of period \(T\) and \(A\) is a constant
\(N\times N\) matrix which is positive (negative) definite (i.e. \(Ax\cdot x\ge\mu\lvert
x\rvert^2\), resp. \(Ax\cdot x\le-\mu\lvert x\rvert^2\), \(\mu>0\)). Use the result proved
above to show that the equation
\[
x'=Ax+h(t)
\tag{2.11}
\]
has a unique \(T\)-periodic solution.

Topic Tags

Gronwall Inequality and Continuous Dependence Linear Systems and Matrix Exponentials Periodic Orbits, Limit Cycles, and Orbital Stability Poincare Maps and Periodic Solutions