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One always needs to allow the frequency to change, otherwise one gets horrible secular terms. In case of resonances one needs additional tricks. A good mathematical book is ''Perturbation methods in nonlinear systems'' by G.E.O. Giacaglia (Springer 2012). He discusses both the traditional Poincare-Linsted method and more advanced methods based on Lie ...


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Instead I found (for the logistical equation and the Lorenz system) that the logarithmic plot of the distance between trajectories is constant at the start This should not happen and I cannot confirm this. Here is the separation of two trajectories I get for the logistic map (averaged over 10000 realisations): And here is the same for the Lorenz ...


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The behavior is approximately constant near the beginning because at very short timescales, the transformation of the phase space induced by the time evolution is basically a deformation where the distances change by a factor of $O(1)$. The middle phase is when the chaos is actually growing and the distances are growing exponentially. At the end, at very ...


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The expression is independent of $\sigma$ and therefore there is no higher harmonics growth etc., only decay of them. I think your exponents have $\sigma$ in them through the $x$ terms, e.g., $$ e^{-n^{2} \alpha \ x} = e^{-n^{2} \alpha \ \bar{x} \ \sigma} = e^{-n^{2} \frac{\sigma}{\Gamma}} $$ One could also find $\sigma$ from $\Gamma = \tfrac{\sigma \ ...


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I would look at this in a slightly different way. Rearranging it: $$ m \ddot{x} = -(a|\dot{x}|+k) x = -k_{eff} x$$ If you look at it that way, it is really a variable, non-linear stiffness $k_{eff}$ that depends on the velocity, rather than a damping that depends on the position. In this respect (assuming $a > 0$), the stiffness coefficient has a lower ...


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is the exit position (or vicinity of the exit) considered an attractor? Sort of. Let’s first consider a single pedestrian who wants to exit the room and whose position in the room is $(x,y)$. Let’s further assume that the exit is located at $(0,0)$. Then we may describe the pedestrian’s position with the following differential equations: $$ \dot{x} = ...


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An attractor is defined in phase space. Phase space is the space of all degrees of freedom of your system. So in your example it cannot be a spatial location such as a room exit. Instead you have to imagine how many parameters describe the motion of one person (a lot), then how many persons there are, multiply the two and you will get the size of phase ...



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