One circle maps under another circle - Dynamical Systems

Question

I've researched a lot trying to do this question. The only way I can think of solving this problem is using a differential equation. But it seems this question requires a discrete map to solve it. Can anyone help?

"Consider the set of points on a circle of radius r about the origin and show that they are mapped under one step of the dynamical system to another circle which you should specify."

You're going to have to tell us what the dynamical system is — Sort of Damocles, Nov 26 '19 at 19:06
Hi, dbx, thank you for your time. " f(x,y)= x^2-y^2+ a; g(x,y)=2xy+b. where a, b are constants and b=0" are the conditions I was given. — Newbee, Nov 26 '19 at 20:11

score 0 · Answer 1 · answered Nov 26 '19 at 22:19

0

Your map is $z^2+(a,b)$ and so the circle of radius $r$ centered at $0$ is taken to the circle of radius $r^2$ centered at $(a,b)$. Alternatively you can write $$ (f(x,y)-a)^2+(g(x,y)-b)^2=(x^2-y^2)^2+(2xy)^2=(x^2+y^2)^2=r^4 $$ to conclude the same.

answered Nov 26 '19 at 22:19

John B

16,854

Thank you, John B. – Newbee Nov 27 '19 at 13:44

One circle maps under another circle - Dynamical Systems

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