Given a system: dx/dt=f (x, y, z, p); dy/dt=g (x, y, z, p); dz/dt=h (x, y, z, p), where p is the set of parameters. The solution of this system can be a stable steady state f, g, h = 0 or stable limit cycle. The solution may also be a quasiperiodic or chaotic. Problem: define functions f, g and h that stable steady state lying outside the area designated by the stable limit cycle, the phase of each of the planes (x, y), (x, z) and (y, z).
See Figs below:
Can f, g and h be defined piecewise? If so, stitch together the required solutions (you only need 2-D to get stable limit cycle in one half-plane and steady state (you mean a stable fixed point?) in the other half-plane - but then, of course, you system will never be chaotic). Just ensure that the two regions are independent invariant sets (with a sepratrix between them).
Alternatively, I think you'll find bifurcations with limit cycles giving birth to fixed points in text books on the subject.
Your figure didn't explain (to me) the various required 2-D phase planes - so maybe my answer does not fit what you have in mind.
Find f, g and h, for which a stable steady state is not within the limit cycle. Limit cycle and a stable steady state are on the same plane phase.
Why dont you like the answer of Michael?
If these functions can be piecewise then one possible solution is
dx/dt = 1 for x=0 \\ 0 for x=3 \\ -y + x(1-x^2-y^2) otherwise
dy/dt = 1 for y=0 \\ 0 for y=3 \\ x + y(1-x^2-y^2) otherwise
dz/dt=0;
This system has an attractive stable limit cycle x^2+y^2=1 and a stable but not attractive equilibrium point (3,3). Is it what you are looking for?
Michael and Stanislav, thanks, but it is not what I am looking for. I am looking for 3 ODEs (continuous), which provide a stable limit cycle and a stable equilibrium point is located on the outside of the cycle. Outside of the stable limit cycle may also be unstable limit cycle.
Summary. On the same plane phase are: stable limit cycle, unstable limit cycle and stable fixed point. All these three objects are placed separately: the fixed point is not inside any limit cycle, similarly, the unstable limit cycle is not inside the stable limit cycle.
Dear Marek,
It seems to me that there is no difference in your case between 2D and 3D systems. However, what do you mean as "stable"? Should these cycle and point be attractive, i.e. assymptotically stable for certain domains?
The second question is - should these f, g, h functions be C0? C1?
Stanislav, see figs 5 and 6 on this paper (in polish, unfortunately). There are PDEs model, but I am looking for this same phenimenom for 3 (or more) ODEs.
2D system can not give this phenomenon.
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