Consequently the solution to an autonomous system is periodic with a closed path, or never passes through the same point twice.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Differential equations. Autonomous systems and stability.
Suppose that the solution (x1(t),y1(t)) is defined for all t and that its trajectory has an apparent self-intersection

x1(a) = x1(a+p) = x0
y1(a) = y1(a+p) = y0

With p>0. with the solution

x2(t) = x1(t+p)
y2(t) = y1(t+p)

And applying the uniqueness theorem, prove:

x1(t+p) = x1(t)
y1(t+p) = y1(t)

for all t. That is, the solution is closed and has a period p.

Consequently the solution to an autonomous system is periodic with a closed path, or never passes through the same point twice.

Please be as clear as possible, and legible. Show and explain all the steps in detail and use definitions if necessary. Thank you very much.

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