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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 ...


18

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 ...


11

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

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 ...


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

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

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

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 ...


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

Comments to the question (v1): Let there be given an $n$-dimensional manifold $M$ with a smooth vector field $X\in \Gamma(TM)$. If $(x^1, \ldots, x^n)$ is some local coordinates on $M$, then the vector field takes the form $$\tag{A} X~=~X^i(x)\frac{\partial}{\partial x^i},$$ and one may study the autonomous first-order ODE $$\tag{B} ...


2

(After possibly introducing more variables) then OP is essentially considering an autonomous system of $n$ coupled 1st order ODEs $$\tag{1} \frac{d\vec{z}(t)}{dt}~=\vec{f}(\vec{z}(t)), \qquad f: U \to \mathbb{R}^n , \qquad U\subseteq \mathbb{R}^n, $$ i.e. without explicit time dependence, so that the system (1) possesses time translation symmetry. OP is ...


2

I think I can remember the derivation for a conservative force field in classical mechanics, wich is a somewhat stronger assumption than pure time-translation invariance. Let $\vec{F}$ be a conservative force field, that is $$ \nabla \times \vec{F} = 0 $$ or alternatively $$ \phi := -\int_\gamma \vec{F} \cdot d\vec{a} $$ does not depend on the path $\gamma$ ...


2

James P. Sethna is one of the leading figure in this area. You can refer to his book without any doubt. http://www.lassp.cornell.edu/sethna/


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

First. Yes, it is possible for the curves in Poincare section to cross. (I am assuming you mean generally). Remember, that Poincare section is a 2D projection of a 3D section of a 4D phase space. Regular, non-chaotic dynamics correspond to the winding of a $T_2$ torus embedded in this 4D space. Sections of such torus could be the couple of closed curves in ...


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, ...



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