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20

Chaotic is not the same as random. A chaotic system is entirely deterministic, while a random system is entirely non-deterministic. Chaotic means that infinitesimally close initial conditions lead to arbitrarily large divergences as the system evolves. But it's impossible, practically speaking, to reproduce the same initial conditions twice. Given ...


16

It has been shown by Eichhorn, Linz and Hänggi in 2000 that the numerical values of Lyapunov exponents are invariant under any invertible variable transform. This is just a reformulation of the fact that they are metric invariant, because the authors presume the norm $|\cdot|$ to be an arbitrary norm in the given coordinates - just it's basic properties such ...


10

The concept of network is very general and can be applied to many physical, biological, neuronal, technological and social phenomenon. Any system with distinguishable individual parts interacting with each other can be described by a type of network. With its applicability to many real world problem, a whole individual research field called network science ...


6

I) In this answer we discuss a systematic approach to linearization and stability analysis. Imagine that the physical system under consideration is described by an autonomous Lagrangian $L=L(q,\dot{q})$ of $n$ generalized coordinates $$\tag{1} q~=~(q^1, \ldots, q^n)~\in~ \mathbb{R}^n.$$ One of the first questions one would like to ask is, if a specific ...


5

Yes. The same system can - at least in many cases - be described by either a stochastic process with memory or by a Markov process. The point is that in order to write it as a Markov process, one must add enough variables encoding the memory. For example, an autoregressive moving average (ARMA) process is defined as a process with memory, but each such ...


4

An harmonic oscillator. When evolving with time, its joint distribution in (p,x) is given by the Boltzman distribution: $e^{-H(p,x)}$, but the energy along a trajectory is constant. Nevertheless if write explicitly the hamiltonian you will find that $e^{-H} = e^{-p^2/2 - x^2/2}$ and although the energy is constant the individual distributions of $x$ and ...


4

It doesn't look that much like a normal distribution to me - particularly on the x axis, the right-hand tail looks heavier than the left, whereas the left one is much longer. But, generally speaking, normal distributions tend to arise when lots of small, independently distributed random numbers (of any distribution) are added together. (The theorem that ...


4

Chaos isn't easy to define precisely, but I'll use the definition from Nonlinear Dynamics and Chaos by S.H. Strogatz to show the features everyone agrees on: Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive dependence on initial conditions. Aperiodic long-term behaviour means there are no fixed points, closed ...


4

The dimension should be 3 or larger. If the dimension is smaller then 3 the existence and uniqueness theorem for differential equations will tell you that functions can't intersect (since you want them to be continuous and differentiable). In 1 dimension this means you can only have movement in one direction In 2 dimensions this means that your value ...


3

No, I would say it is wrong to immediately conclude that there is no scale just because the variance diverges. Only functions of $x$ of the form $x^n$, a power law, have a chance to be considered scale-free; none of these functions may be considered a probability distribution because the integral diverges. Any other function – and therefore any normalizable ...


3

According to Nonlinear Dynamics and Chaos by Steven Strogatz The requirements for chaos are: Deterministic system (only one future for each state) Irregular spatial, temporal, or spatiotemporal patterns (a qualitative feature) A positive maximum Lyapunov exponent. 3) is pretty much the quantitative standard in journals of chaos, assuming you meet the ...


3

What you are seeing are artifacts caused by the hard-wall type kernel you are using. These patterns consist of horizontal and vertical streaks with a definite periodicity, which gets more pronounced as you make the averaging box bigger. The streaks occur because the Fourier transform of a hard-wall box has zeros at certain wavenumbers. To get rid of them, ...


3

The roll off deviation appears to mostly be due to difficulties with accurate measurements at low magnitudes. In order to preserve the GR law you'd need to exhaustively record all earthquake measurements below the roll off magnitude and this is largely infeasible. A good example to look at (figure 3.1) is the difference between the Sumatra 2004 and Kobe ...


3

What's $V_{ab}$? Well, $V_{ab}$ is the "symmetric, positive definite potential energy matrix". Ok lol I'm trolling here, but as the name suggests, $V_{ab}$ describes the strength of the (linearized) interaction between particles $a$ and $b$. To be precise, it is the second derivative of the potential energy function of the system with respect to $u_a$ and ...


3

Perhaps a better question to ask is: why is a single pendulum non-chaotic? Almost all real systems are chaotic at least to some extent; the fact that we can write out the solution for a single pendulum for all points in time is really quite peculiar, and only true because it is a highly simplified system. The reason these non-chaotic systems are so prevalent ...


3

It a widely known and experimentally useful fact in nuclear and particle physics that the position and momentum distributions of bound systems are related to one another by a Fourier transform. Is the system you are inspecting bound? The tails in the data that Nathaniel notes suggest that it is not fully bound, which means the Fourier relationship between ...


3

This answer contains some additional resources that may be useful. Please note that answers which simply list resources but provide no details are strongly discouraged by the site's policy on resource recommendation questions. This answer is left here to contain additional links that do not yet have commentary. Uriel Frisch, Turbulence: the legacy of ...


3

The critical point of a general statistical system is a point in the space parameterized by intensive quantities, especially temperature and pressure, at which there exist no boundaries between two different phases of the material even though the boundaries exist at an infinitesimally nearby point. It's the end of a co-existence curve for two phases. ...


3

$f=-kx$ is stable whereas $f=kx$ is unstable. You can usually rewrite a matrix $A$ as $A=PDP^{-1}$ where $P$ is a matrix of eigenvectors and $D$ is a diagonal matrix of eigenvalues. If $F=Ax$, then by the above, $(P^{-1}F)=D(P^{-1}x)$. Now you have $n$ independent equations exactly of the form $f=kx$ or $f=-kx$. If any one of them is like $f=kx$, the ...


3

First, you need a deterministic dynamical system. By deterministic it means that the state of the system is univocally determined at each time, ie. at each time you have one and only one possible state. In the counterpart are stochastic systems where, instead, the state of the system is determined by a distribution of possible states and is this distribution ...


2

1st question: Yes all systems have areas of periodic (or almost-periodic, or isolated periodic) motion. For integrable systems this is the expected. But in fact the existence of isolated periodic solutions is a form of non-integrability a well (related to 2nd question as well). For the exact meaning of isolated periodic solution check the link on ...


2

Let us look at a one-dimensional example: Recall that $f(x) = \dot{x}$, so $f$ encodes the time evolution of $x$. If $f < 0$, then $x$ will move to the left. If $f > 0$, then $x$ will move to the right. If $f = 0$, $x$ will not move at all, this is why $f(x_0) = 0$ is the equilibrium condition. Now, look what happens if you perturb the equilibria ...


2

If x(t) is a random process it is quite unlikely that the derivative xdot(t) exists. So your description looks somewhat problematic. It seems that you have a Wiener process (= random walk, Brownian motion). See http://en.wikipedia.org/wiki/Wiener_process Here the changes in x are Gaussian and uncorrelated with x itself. Then x itself also follows a ...


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, ...


2

I'm going to interpret what you're asking as follows: The moon orbits the planet, the planet orbits Sun #1, and Sun #1 forms a binary star system with Sun #2. The planet is close enough to Sun #1 that the gravitational effects from Sun #2 are small. I believe this is a stable arrangement, particularly if Sun #2 is very far away. It's quite similar to the ...


2

In general, it can be hard to tell if a given set of equations of motion (eom) are part of a (possibly larger) set of eom that can be put on Hamiltonian (or on Lagrangian) form. Specifically, OP asks about the Kuramoto model with eom $$\tag{1} \dot{\theta}_j -\omega_j ~=~\frac{K}{N}\sum_{k=1}^N\sin(\theta_k-\theta_j) ~\equiv~ K ~{\rm Im} \left( ...


2

As Michael Brown asked for an elaboration, here's a short introduction into the geometry of Hamiltonian mechanics: While the traditional formulation works with special sets of coordinates (the 'canonical' ones), the differential-geometric approach de-emphasizes these and instead introduces the symplectic form as the characteristic structure of Hamiltonian ...


1

A standard/famous book on dynamical systems, which is very good about nonlinear stability is "Nonlinear Dynamics And Chaos" by S. H. Strogatz.


1

No, or at least not in the sense the phrase "butterfly effect" is normally used. Well, possibly, but only if the critical system is chaotic. The phrase is normally applied to systems that show chaotic behaviour. In such systems the trajectory of the system is very sensitive to the starting point i.e. if you take two points very close together in phase space ...



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