Timeline for (In)stability of a two-dimensional dynamical system
Current License: CC BY-SA 4.0
43 events
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| Apr 9, 2022 at 8:09 | history | edited | Denis Serre | CC BY-SA 4.0 |
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| Apr 9, 2022 at 8:00 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
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| Jul 28, 2018 at 20:12 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 26, 2018 at 2:06 | answer | added | David Hughes | timeline score: 1 | |
| Jul 23, 2018 at 2:56 | comment | added | Piyush Grover | @user539887 I am not sure if there is an exact theorem, but you might find book "Galois theory of linear differential equations" useful. | |
| Jul 22, 2018 at 23:12 | comment | added | David Hughes | Floquet theory applies to linear systems with periodic matrices, and can be sometimes used to deduce the stability of the origin. In this case all we can say is that the Floquet exponents sum to zero, but that does not decisively rule out stability. I was thinking a perturbation approach might work. | |
| Jul 21, 2018 at 9:51 | comment | added | user539887 | @PiyushGrover Is that a statement in differential algebra? Could you give some references? | |
| Jul 21, 2018 at 3:27 | comment | added | Piyush Grover | A generic Floquet problem is not solvable analytically. | |
| Jul 21, 2018 at 1:11 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 23:56 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 16:19 | comment | added | Ludwig | @user539887: You are absolutely right: matrices do not commute! (I removed my wrong approach...) Anyways, do you have some ideas to tackle this problem? | |
| Jul 19, 2018 at 16:17 | history | edited | Ludwig | CC BY-SA 4.0 |
Removed wrong approach
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| Jul 19, 2018 at 12:54 | comment | added | user539887 | I think that in your edits the matrix $$\begin{bmatrix}e^t&0\\ 0&e^{\lambda t}\end{bmatrix}$$ to the left of $x$ is missing. Further, how do you have that a fundamental matrix is given by $\exp(\int_0^t A(\tau)\,d\tau)$? The matrices $$\begin{bmatrix}-1&e^{(1-\gamma)t}\\ e^{(\gamma-1)t}&-1\end{bmatrix}$$ don't commute for $t_1\ne t_2$ unless $\lambda=1$. Of course, commutation is not a necessary condition. | |
| Jul 19, 2018 at 3:52 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 3:40 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 3:03 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 2:41 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 1:16 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 19, 2018 at 1:06 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 18, 2018 at 17:03 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 18, 2018 at 5:47 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 18, 2018 at 4:55 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 16, 2018 at 3:12 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 14, 2018 at 22:00 | history | edited | Ludwig | CC BY-SA 4.0 |
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| Jul 14, 2018 at 19:37 | history | edited | Konstantinos Kanakoglou |
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| Jul 14, 2018 at 19:15 | comment | added | Ludwig | @KonstantinosKanakoglou: I've edited the question adding the Mathematica code that I've used in my simulations. | |
| Jul 14, 2018 at 19:14 | history | edited | Ludwig | CC BY-SA 4.0 |
Added Mathematica code
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| Jul 14, 2018 at 19:07 | comment | added | Konstantinos Kanakoglou | Interesting! could you add some details on the numerical simulations you've used (some code for example) ? | |
| Jul 14, 2018 at 19:04 | history | asked | Ludwig | CC BY-SA 4.0 |