I mean a 3D chaotic continuous system in which any point on a specific curve (or a surface) is equilibrium and no other equilibrium exists beside that.
Any idea about characteristics of such systems?
That system may be chaotic, it depend on the stability of the equilibria. It's may be occur where you have unstable points.
Take a chaotic periodically forced nonlinear 2d system, make it an autonomous 3d system in R^2 x S^1 and multiply the flow with a scalar function that is zero on a curve or surface which lies outside of the chaotic attractor but nowhere else.
Dear Ralf
I said in the question "no other equilibrium exists beside that".
Indeed, why would there be another equilibrium, if there is none in the periodically forced system? By construction, only the points where the scalar function is zero are equilibrium points.
Thank you. However I hoped I could find a published system with that property. There are many 3D chaotic systems with no equilibrium. Using each one of them, we can obtain a system with curve equilibrium. I wanted something different.
It's an interesting question although I'm confused as to why you used the word "only". If you have a one or two dimensional curve/surface of fixed points, that is "a lot" of fixed points in the following sense. A one or two dimensional curve or surface of fixed points is non-generic. Consider a 3D flow given by: dx/dt = f(x,y,z), dy/dt=g(x,y,z) and dz/dt=h(x,y,z) . The equilibria (x*,y*,z*) obey: f(x*,y*,z*)=g(x*,y*,z*)=h(x*,y*,z*)=0: 3 equations for 3 unknowns. Generically the fixed points will be discrete points. Curves and surfaces would imply that there were fewer equations than unknowns, i.e. that the 3 functions are not independent. That might be enough to rule out the possibility of chaos. For example if f = -g - h then x+y+z = constant -> 2 dimensional dynamics -> no chaos. I think that more generally the curve or surface you postulate probably implies the existence of one or two conserved quantities which implies that the dynamics are lower than 3-dimensional?
Dear Professor Pearson,
In some chaotic system if you multiply all the equations in a new equation (this new equation can be zero in a curve or surface), since there will be no change in dx/dy, dx/dz, dy/dz if we start from a point in the basin of attraction of the original chaotic system, we converge to its strange attractor. That’s why I said “only”. However this trick can still be used since there are chaotic systems with no equilibria (see “elementary quadratic flows with no equilibria”). However I meant none of them. Recently we (I, Chunbiao Li, and Professor J.C. Sprott) have found a family of such systems (systems without trick). I checked famous chaotic systems and all of them had countable fixed points. I asked this question to find out if there are some cases which I have missed.
Thank you for your interest.
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