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11

Not all nonlinear systems are chaotic. However a chaotic system is necessarily nonlinear. There doesn't exists a definition for chaos but using the one given by Strogatz, ref 1: Chaos is aperiodic long-termed behavior in a deterministic system that exhibits sensitive dependence on initial conditions. Like explained in the text: aperiodic long-termed ...


7

If you want to generalize a potential to a class that's broader than the simple $\frac12 k_2 x^2$, it is tempting as a first step to include a small perturbation of the form $\frac13k_3x^3$. Unfortunately, this drastically changes the structure of the potential, because it becomes unbounded from below. Thus, you might get a slightly perturbed behaviour ...


6

Numerical simulations are not always meaningful, as chaos theory belongs to the large subject of dynamical systems theory. Although the definitions differ, chaos generally occurs in three contexts: Sensitive dependence on initial conditions (SDIC). The set is topologically transitive. Periodic points are dense in the set. Think of two particles having ...


4

It is an interesting problem. Usually you would find the infinitesimal oscillations by setting $p_i = b + \delta p_i$, $q_i = a + \delta q_i$ and expanding the Hamiltonian to second order in the $\delta p_i$, $\delta q_i$. Here though, this doesn't work as you just get the same Hamiltonian, $$ H = \sum_{i,j} M_{ij} \sqrt{(\delta p_i - \delta p_j)^2 + (\delta ...


4

The interplay of Hamiltonian and Lagrangian theory is based on the following general identities, where $L$ is the Lagrangian function of the system, $$\dot{q}^k = \frac{\partial H}{\partial p_k}\:,\qquad(1)$$ $$\frac{\partial L}{\partial q^k} = -\frac{\partial H}{\partial q^k}\:.\qquad(2)$$ Above, the RH sides are functions of $t,q,p$ whereas the LH sides ...


4

Since the rate of change of $ x $ is the same as the rate of change of $y $ you really only a single equation of with one variable. We write, \begin{equation} x = y + c \end{equation} where the constant $ c $ is determined by your initial conditions, \begin{equation} c = x (0) - y (0) \end{equation} (in your case it is the difference between the ...


4

Three different points of views on essentially the same thing: Chaotic systems are not only sensitive to numerical errors, but also to any other small perturbations, such as dynamical noise, which may simulate real conditions. Though tiny perturbations affect the detailled, microscopic future of a system, its qualitative dynamics is unaffected. And the ...


3

The other answers address how to solve this analytically, but I like numerical solutions to things so here goes: $$\frac{d}{dt} \begin{bmatrix}x \\ y \end{bmatrix} = -\begin{bmatrix}kxy\\kxy\end{bmatrix}$$ which can be solved using any number of numerical methods. For simplicity, we can take the second-order Runge-Kutta method where $i$ is the time index. ...


3

Hints: Conclude that $y-x=c$ is a constant. Use separation of variables $-k\int \!\mathrm{d}t= \int \!\frac{\mathrm{d}x}{x(x+c)}$.


3

There is a normalized form, though it's properly called the dimensionless Euler equations. The way to do it is define: scale time $t_0$ scale density $\rho_0$ scale length $L_0$ and then derive the scales from these: $$ v_0 = \frac{L_0}{t_0},\quad p_0=\rho_0v_0^2 $$ NB: it is possible to use other combinations, but I find that these are often the ...


2

1D Burger's equation is not meant to model a physical phenomenon. Rather, it is a simplification of homogeneous incompressible Navier-Stokes equations that preserves (some of) its mathematical structure: the non-linear convection term and the second order derivative of viscous forces. It was initially intended as a useful simplification to try to ...


2

I can think of a method, although it may require to compute $\mathbf{F}_0$ for a very large number of test points. It is based on Gauss's law for gravity $$ \frac1{m_0}\oint_{S} \mathbf{F}\cdot\mathrm d\mathbf S=-4\pi GM_S$$ where $S$ is a closed surface and $M_S$ is the total mass contained inside it. So the idea would be to use some numerical scheme to ...


2

You might try measuring the temporal coherence of the laser. (This came to mind 'cause it's closely related to the PhD work I would have done if I didn't quit with an ABD :-( ). Basically, you split the beam with, e.g., a 50-50 splitter, and recombine one way or another to get an intereference pattern. Then use an "optical trombone" (the meaning should ...


2

One of the simplest measurements you can do is to characterize the spatial mode coming out of the laser. This can be done with a simple photodiode and a scanning slit (optical chopper or razor blade mounted on a movable stage). I wrote a brief note on making such a measurement here. This isn't the most interesting of measurements if you aren't going to ...


2

However, what if the time series is multi-dimensional, indeed of the same dimension as the phase space, to begin with? Well, how would you know that your time series is of the same dimension as the phase space? Usually, because you already know the dynamical equations for your system (as for your pendulum). If you observe a real-life complex system, ...


1

A dynamical system is a system that evolves by a rule over time. As opposed to modeling a system with a PDE, the system is modeled by some type of iterated function, $f^t(x)$. Smooth implies that the function being iterated is differentiable, so we are not talking about an iterated system like cellular automata.


1

As a check, for your linear stability matrix I get: $$A = {\left. {\left( {\matrix{ {3{x^2} + a} & 1 \cr 1 & { - 1} \cr } } \right)} \right|_{(x* = 0,y* = 0)}} = \left( {\matrix{ a & 1 \cr 1 & { - 1} \cr } } \right)$$ which is presumably what you have as well. By the way, did you switch notation from, $a \to \alpha $? ...


1

Clarifications and addition: It's true that not all nonlinear systems are chaotic, but that all chaotic systems are nonlinear (or infinite-dimensional linear). The sensitivity to initial conditions is an important point and the commenter raises a good question. Consider the Lorenz system in a non-chaotic parameter regime (or for that matter, any stable ...


1

Usually averages of any total time derivatives vanish. Easiest way to see this is to use the fact that steady state time averages and ensemble averages should be the same if the system is ergodic....therefore $$ <dA/dt>= \lim_{T \to \infty} (1/T) \int_0^T dt dA/dt = \lim_{T \to \infty} (1/T) [A(T)-A(0)] =0 $$ unless $A(t)$ is unbounded.


1

So I was wondering whether it is still true even when the system is dissipative like a damped harmonic oscillator? It is true if the dissipative system is Hamiltonian, i.e. if the dissipative behaviour can be described by time-dependent Hamiltonian. For example, one oscillator connected to million oscillators can be described by the Hamiltonian $$ H = ...



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