Timeline for Noninvariance for a specific nonlinear oscillator
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 26, 2015 at 12:10 | comment | added | user45183 | @AliTaghavi Thanks for your answer. I am not sure what you mean exactly with counter example. I am not trying to find one example which satisfies the property but I am trying to show that all solutions starting with $x_2(t) + z_2(t) = 0$ (starting near the origin) will have $x_2(t) + z_2(t) \ne 0$ at some point in time. | |
| Oct 26, 2015 at 11:34 | comment | added | Ali Taghavi | @seno44 I would like to add a related question: is there a periodic orbit $\gamma$ with period $T$ such that $\gamma$ start from $(0,y))\;\;y>0$ and arrive at $(0,-y)$ at time $T/2$?If there is a periodic orbit with this property, it would be a counter example to your question. | |
| Oct 26, 2015 at 11:05 | answer | added | Frits Veerman | timeline score: 2 | |
| Oct 24, 2015 at 22:38 | comment | added | user45183 | Thanks for your efforts. I know that the potential is given by $U(x) = \frac{1}{2}x_2^2 + 2 x_1^2 - \frac{1}{3} x_1^3$. The separatrix I am actually not interested in, only the periodic orbits close to the origin. | |
| Oct 24, 2015 at 21:32 | comment | added | Dr. Wolfgang Hintze | @ seno44: your system describes an anharmonic oscillator (x,y = dx/dt) with the potential $U = 2 x^2 - \frac{1}{3} x^3$ which can be treated with standard methods. Energy conservation is an elliptic curve. The separatrix between finite and unbounded orbits is given by $y^2=\frac{2 x^3}{3}-4 x^2+\frac{64}{3}$ | |
| Oct 24, 2015 at 20:03 | answer | added | Dr. Wolfgang Hintze | timeline score: 1 | |
| Oct 24, 2015 at 3:43 | comment | added | Piyush Grover | One idea: work in extended phase space of two copies of the original system (i.e. state space being $x_1,x_2,z_1,z_2$. Then do a coordinate transformation $y_1=x_2+z_2, y_2=x_2-z_2$ and look at fixed points of this 4D system. | |
| Oct 23, 2015 at 18:39 | history | asked | user45183 | CC BY-SA 3.0 |