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Suppose I have the following system:

$dx/dt=f_1(x,y,z)$

$dy/dt=f_2(x,y,z)$

$dz/dt=f_3(x,y,z)$

Now it is given that $x(t)\leq K_1$,$y(t)\leq K_2$,$z(t)\leq K_3$, i.e the solutions are bounded in a region R. Also it is known that that $\frac{\partial }{\partial x}gf_1+\frac{\partial }{\partial y}gf_2+\frac{\partial }{\partial z}gf_3$ maintains same sign in the region R, where $g$ is a positive function.Using Dulac's criterion we can conclude that the system has no closed orbits. Can we conclude that the system is globally asymptotic stable? If yes, please provide the proof or else a counterexample. Global stability definition can be found here on the 2nd page: https://stanford.edu/class/ee363/lectures/lyap.pdf

Thanks in advance.

Germain
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    Can we stop at applying Dulac-Bendixson theorem to 3d systems? It works only for 2d systems. – Evgeny Sep 22 '17 at 23:04
  • There exists extension version of Dulac's negative criterion. Can we get any help from there?I need an answer for this urgently so putting this for bounty. – Germain Oct 03 '17 at 16:04
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    But you are omitting very important part of statement from this paper! I quote it here: "We show in this note that if system has $p$ independent first integrals, the Bendixson criterion ${\rm div}, f \neq 0$ implies the nonexistence of certain invariant $n-p-1$ dimensional objects of system on each non-degenerate level set of the first integrals". When you have no first integrals ($p=0$) the least this theorem could do for you is to prove that for example there is no invariant tori. – Evgeny Oct 03 '17 at 17:54
  • This is a part of my research problem. It is a 3D system and am able to prove it locally stable, uniformly bounded and satisfies Dulac's negative criterion. I am looking for a proof of global stability. Can you help with any idea other than constructing a Lyapunov function to prove the global stability part? Thanks in advance. – Germain Oct 04 '17 at 12:09
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    So you start with a 3D system which has equilibrium at the origin which is locally stable. Also you have some invariant bounded domain in which generalized Dulac-Bendixson criterion holds. Do you have a particular system? Or is this a theoretical question? – Evgeny Oct 04 '17 at 14:25
  • numerically I have already checked the system is globally stable using MATLAB, now I need to prove it theoretically. Thanks for your patience. – Germain Oct 04 '17 at 14:48
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    It's really hard to say for an abstract system how to do this. Maybe your system is special and it is possible to fiddle and find Lyapunov function. The problem is that in case of 3d systems beyond equilibria, limit cycles and tori strange attractors are also possible, and it's not a trivial matter. For sure we know that you don't have invariant tori :) do you know are there any other equilibria? – Evgeny Oct 05 '17 at 05:02
  • yes there is annother equlibria which is unstable for the same set of parameter values. The second equlibria is behaving globally stable in MATLAB, I need to prove this theoretically. Please provide some idea with any other possible way of proving global stability other than constructing Lyapunov function. Thanks again! :) – Germain Oct 06 '17 at 03:08
  • You should write out your system. Without further details, it is almost impossible to say anything about your question. – MrYouMath Oct 06 '17 at 09:21

1 Answers1

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If your system has the following form

$$\dot{\boldsymbol{x}}(t)=\boldsymbol{P}\boldsymbol{x}+\boldsymbol{q}f(\boldsymbol{r}^T\boldsymbol{x}),$$

in which $\boldsymbol{x}\in \mathbb{R^3}$, $\boldsymbol{P}$ is a constant $3 \times 3$ matrix, $\boldsymbol{q}$ & $\boldsymbol{r}$ are constant $3$-dimensional vectors and $f$ is a differentiable scalar function with $f(0)=0$ such that

$$k_1 < f(\boldsymbol{r}^T\boldsymbol{x}) < k_2.$$

Then the system is stable in the large (i.e. a unique stationary point is a global attractor) if all systems with $f(\boldsymbol{r}^T\boldsymbol{x})=k\boldsymbol{r}^T\boldsymbol{x},$ $k\in (k_1,k_2)$ are asymptotically stable.

This statement is called as Kalman's conjecture, it is only valid for systems with $n\leq 3$ for continuous systems. For discrete systems it is only valid for $n\leq 1$.

MrYouMath
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