Timeline for Stability analysis of equilibrium point of non-linear ODE system with Jacobian going to infinity
Current License: CC BY-SA 4.0
15 events
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| Jan 31, 2023 at 20:18 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Minor formatting
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| Jan 25, 2023 at 4:14 | comment | added | fedja | I guess I figured it out by now and it is exactly as I supposed. It is too late to post today and I'll be pretty busy tomorrow, but I'll try to make an update soon. Note also that when you suggested to replace $T_a/(T_a+c_{T_a})$ by the square root expression in the posted model, you didn't notice that that ratio appears only in the product with another variable ($M_{un}$ in that case). If that variable is a part of your equilibrium at $0$ (so you have $X_k\sqrt{X_j}$, not $\sqrt{X_j}$ alone), then it can be disregarded as a "higher order term" and the usual linear stability analysis applies. | |
| Jan 25, 2023 at 0:31 | comment | added | fedja | @PiyushGrover You are not allowed to start at $0$ in such models. The question is about the solutions remaining in the "positive octant" for all times where everything is OK. But this requires quite special structure of the system (see my model description below). | |
| Jan 24, 2023 at 22:47 | comment | added | Piyush Grover | Doesn't this imply the function is not Lipschitz and hence even uniqueness is not guaranteed ? | |
| Jan 24, 2023 at 22:45 | answer | added | fedja | timeline score: 3 | |
| Jan 24, 2023 at 20:48 | history | edited | LSpice | CC BY-SA 4.0 |
Capitalise title; tidying
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| Jan 24, 2023 at 20:32 | comment | added | fedja | I see. That clarifies a lot about the system structure. Let me think of it now. By degeneracy I meant something like that in the system $\dot C=-AC, \dot A=-A$ when there is no or degenerate linear part at $(0,0)$, so the Jacobian test becomes inconclusive. | |
| Jan 24, 2023 at 19:38 | comment | added | Omega | I've added an example into my question :) What type of degeneracy we are talking about here? @fedja | |
| Jan 24, 2023 at 19:35 | history | edited | Omega | CC BY-SA 4.0 |
added 539 characters in body
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| Jan 24, 2023 at 12:29 | comment | added | fedja | Sorry for my being unclear: I meant a minimal example of an actual system of ODE's that has all those features (I suspect that in addition to having an unbounded Jacobian, your system is also degenerate at the origin in a certain way (like those chemical system where the RHS is the sum of products of variables) at least as long as $X_j$ is concerned. And if the RHS are always all positive, what stability at $0$ can we be talking about? | |
| Jan 24, 2023 at 11:42 | comment | added | Omega | Let's imagine a simple chemical (or biological) system in which there are catalysts and inhibitors. It is constructed in such a way that $X_i$ never becomes negative, since obviously we are dealing with concentrations and densities. So, $X(t)_j^{0.5}+b_j^{0.5} > 0$ for all t. @fedja | |
| Jan 24, 2023 at 1:04 | comment | added | fedja | Can you design a simple model that reflects the main features of your system? The most interesting part is the equation for $\dot X_j$: there should be something that prevents it from hitting $0$ and I want to see what that is before saying anything. | |
| Jan 22, 2023 at 17:28 | review | Close votes | |||
| Feb 4, 2023 at 19:52 | |||||
| S Jan 22, 2023 at 10:42 | review | First questions | |||
| Jan 22, 2023 at 13:12 | |||||
| S Jan 22, 2023 at 10:42 | history | asked | Omega | CC BY-SA 4.0 |