All Questions
Tagged with complexity or complex-systems
405 questions
9
votes
2
answers
324
views
What is the reason for power laws appearing at phase transitions?
Let's consider a system with order parameter $\rho$ (e.g. density in liquid-gas transition, magnetization in Ising model) and control parameter $\tau$ (usually the normalized temperature, but it could ...
- 294
2
votes
0
answers
29
views
Find closed orbit problem (Strogatz 8.3.2) [closed]
I'm having trouble solving this excercise from Strogatz
Consider the following system for a chemical oscilator:
$$
\dot x= a -x+x^2y
$$
$$
\dot y=b -x^2y
$$
Where $a,b>0$ are parameters and $x,y\...
- 97
1
vote
0
answers
73
views
Given an infinite amount of time, will every possible combination of matter pop into existence?
Apparently it is true that when the universe is in the state of heat death, quantum fluctuations will eventually produce every combination of matter, no matter how unlikely, given an infinite amount ...
- 137
5
votes
3
answers
800
views
How can bounded orbits diverge arbitrarily far at late times?
While there are many definitions of chaotic dynamics, many commonly used ones require a positive Lyapunov exponent, which is defined as
$$\lambda := \lim_{t \to \infty} \lim_{\delta{\bf Z}_0 \to {\bf ...
- 49.4k
-3
votes
1
answer
123
views
Lyapunov is wrong - got unstable on a stable system [closed]
I'm angry with the Lyapunov stability criteria. Consider this system:
Here, $u$ is the input and $x_1$, $x_2$ are my state variables. Now, solve for the transference of the system, defining my output ...
- 337
1
vote
2
answers
128
views
Is it possible to define a universal formula for chaos?
I've been working on a chaos project. I have noticed that there are several formulas to find the behavior of chaos, for example:
The logistic map is a simple equation that exhibits chaotic behavior ...
- 37
0
votes
1
answer
60
views
Density-Dependent Diffusion Dispersion Relation
I'm studying a model which incorporates the following PDE
$$\frac{\partial u}{\partial t}=uf(u)+D(1-\alpha f(u))\frac{d^{2}u}{dx^{2}}-D\alpha\frac{df(u)}{dx}\frac{du}{dx}$$
With zero flux boundary ...
- 93
8
votes
1
answer
341
views
How are far from equilibrium systems studied analytically?
I've read about stuff having to do with complex systems where some pretty wacky stuff happens, mostly involving "phase changes", which as I understand don't really have much to do with ...
- 93
0
votes
0
answers
20
views
How do you measure the criticality of the phase state of a dynamical system?
Does anyone know how to measure the 'critical-ness' of a dynamical system in which it's configured to allow 2nd order phase transitions? For context, I'm an undergrad ML researcher that is doing ...
- 1
3
votes
1
answer
43
views
Self-confinement of $N$-body gravitational systems
Consider $N=2$ point particles (each having unit mass) interacting via Newtonian gravity in the usual 3-dimensional space. There is a simple criterion to assess whether the system is bounded or not: ...
- 5,244
3
votes
1
answer
41
views
Shadowing Lemma and Numerical Integration
Shadowing lemma tells that any pseudo-trajectory (numerically integrated trajectory) from some initial condition $x_0$ is the exact trajectory of a different initial condition $x_1$.
Q: Is this ...
- 61
1
vote
0
answers
108
views
Conserved Noether charge in complex plane
I'm having some problems in understanding charge conservation. I'm reading Polchinski "Introduction to string theory" cap. 2. Here, we are talking about conformal field theory. Specifically, ...
1
vote
0
answers
78
views
Symmetry and integrability in classical Hamiltonian
I am trying to understand the behaviour of an Hamiltonian system I'm simulating. I will give a quick context setting. The system is defined as
$$
\mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}...
- 61
0
votes
0
answers
50
views
Can all solutions from any possible dynamical systems be formulated as combinations of harmonic oscillators?
In my understanding, all physical oscillators are unit vectors of a Hilbert space that is represented from the group $SU(3)\times SU(2)\times U(1)$. Every dynamical system operates under this group (...
- 929
-2
votes
1
answer
113
views
Advanced math courses for theoretical physics students [closed]
Theoretical physics studies concerning statistical mechanics, dynamical systems and analytical classical mechanics all require working knowledge of mathematical concepts and theories (e.g. manifolds, ...
2
votes
0
answers
87
views
Connections between the Classical and Quantum Three-body Problem?
If mathematicians somehow find an exact analytical solution to the three-body problem, would that help solve the Schrödinger equation exactly and analytically for the helium atom?
And more generally, ...
- 129
0
votes
1
answer
58
views
Strogatz's condition on definition of energy
In, Nonlinear Dynamics And Chaos, 2nd edition page 160, by Steven H. Strogatz, he writes
Let’s be a bit more general and precise. Given a system $$\dot x =f(x),$$ a conserved quantity is a real-...
3
votes
2
answers
206
views
Epidemic spreading model
I'm studying a model in the field of complex systems regarding the epidemic spreading. The model is the susceptible-infected model, i.e., there is a population of N subjects and each of them can ...
- 951
6
votes
6
answers
394
views
How can I formalize better this proof that angular momentum is conserved for a small impulse?
The book I am studying is discussing Lagrange stability of circular orbits, which assumes fixed angular momentum ${\bf L}$, hence in an introductory paragraph explains why, when studying stability of ...
- 114
1
vote
1
answer
67
views
On complex impedance representation and Riemann surfaces
We know that a complex number, $z=re^{i\phi}$, can be represented with infinitely many phases, $\phi+2\pi n$, for integer $n$, as can be easily seen from the equivalent picture of a vector on the ...
- 632
1
vote
0
answers
92
views
Holonomic constraints as a limit of the motion under potential
In Mathematical Methods of Classical Mechanics, Arnold states the following theorem without proof in pages 75-76:
Let $\gamma$ be a smooth plane curve, and let $q_1, q_2$ be local coordinates where
$...
- 133
0
votes
0
answers
25
views
Average resultant length $L$ of an active string initially of length $L_\mathrm{0}$ after a single point mutation
I stumbled upon this research article, where they define a digital organism as an abstract minimal model of an evolving predator-prey system as follows:
An organism is defined via its genome of fixed ...
0
votes
0
answers
74
views
How to find the stability of time dependent Lyapunov equation?
After linearization of the nonlinear equations, I want to find the covariance matrix $v$ through the numerical solution of time dependent Lyapunov equation, $$dv/dt=a*v + v*a'+ d,$$ where $a$ is my ...
- 31
0
votes
1
answer
35
views
Question regarding dynamic modeling of cable driven robot
I am studying rigid body dynamics and am currently exploring the paper "Analytic determination of wrench closure workspace of spatial cable driven parallel mechanisms". In the paper, the ...
- 13
3
votes
0
answers
101
views
Classical mechanics: Hamiltonian perturbation theory. What if the perturbing parameter is < 0?
In Hamiltonian Perturbation theory, we have a Hamiltonian of the form $$H(q,p) = H_0(q,p) + \lambda H_1(q,p).$$
One proceeds by expanding the equations of motion in powers of $\lambda$, assuming $\...
0
votes
0
answers
164
views
Lyapunov Exponent for Double Pendulum
I want to calculate the Lyapunov Exponent for a double pendulum, with a small change in the initial angle. In this study, the authors used the formula $\frac{1}{t}{ln(\frac{d}{d_0})}$ as $t$ tends to ...
- 136
0
votes
1
answer
88
views
Are there any "linear" lagrangian systems of interest for which an analytic solution is not obvious?
Out of curiosity, I am interested in Lagrangian dynamical systems that can be expressed in a "linear" manner. By this, I mean that their Lagrangian can be expressed, quadratically, as
$$L = \...
- 175
2
votes
1
answer
64
views
Meaning of Gibbs average
This might sound a very amateur question but I couldn't find the answer anywhere. Simply, what is the meaning of Gibbs average? I've came across this term on the paper: Transmission of Information ...
- 61
3
votes
3
answers
304
views
Why is the brain so "efficient"? [closed]
By efficient I mean, that why is the brain able to learn tasks, for example driving, at a much less energy cost than Machine learning models?
From a quick google search it seems like it took about 55 ...
- 165
2
votes
0
answers
39
views
Measuring "complexity"
A recently popular idea in the quantum theory of black holes is that there is an isomorphism between the interior state's volume and the computational complexity of the CFT dual (in an AdS/CFT setting)...
1
vote
0
answers
317
views
Computing the system dynamic complexity using the entropy
In the thesis "Structural Complexity and its Implications for Design of Cyber-Physical Systems," the author, K. Sinha, defines the $C$ complexity of a dynamical system in the following way:
\...
- 186
1
vote
0
answers
48
views
Two-body problem + shield
Let us consider two point charges, one positive, one negative, interacting via Coulomb force.
In the absence of any other force, this system constitutes an elementary example of two-body problem, and ...
- 1,252
2
votes
1
answer
389
views
Binder cumulant method for non-Gaussian distributions
In the Ising model, we know that the order parameter $m$ has a Gaussian distribution for temperatures below the critical point. Measuring the exact point where this phase transition takes place was ...
0
votes
1
answer
78
views
Is it generally possible to determine how long it takes for a system to reach its stationary state, without simulation?
Let's say I'm dealing with the heat equation with some initial and boundary conditions, for example example our system can be a $1 \times 1$ metallic plate. Assuming that the initial temperature ...
- 3,430
6
votes
1
answer
84
views
Perturbations of an integrable system with no resonant tori
Suppose I have a Hamiltonian $H_0$ which is just a collection of $N$ non-interacting harmonic oscillators. Written in action-angle coordinates $(J_i, \theta_i)$ we have $H_0 = \sum_{i=1}^N \omega_i ...
- 8,917
13
votes
6
answers
1k
views
(A modification to) Jon Pérez Laraudogoita’s "Beautiful Supertask" — What assumptions of Noether's theorem fail?
I am curious about the following (physically unrealizable) scenario involving a supertask described here: https://plato.stanford.edu/entries/spacetime-supertasks/#ClasMechSupe. The original paper is ...
- 8,776
18
votes
3
answers
22k
views
Why did these algae grow like this in the pool? Are these curves the gravitational equivalents of the bell curve?
My friend sent me these pictures of a pool that has been abandoned for a long time, and we are curious about the reason behind the peculiar growth of algae in this pattern. The needle-like towers of ...
- 318
8
votes
3
answers
888
views
Why do good materials operate in non-equilibirium conditions?
If we look at the majority of useful or industrial materials surrounding us, like metallic alloys, glasses, ceramics, or plastics, it is often the case that these materials went through really hard ...
- 905
1
vote
1
answer
236
views
Can the conservation law be extended to the 2d Burgers equation?
I know that for the 1d inviscid Burgers' equation of the form $$\frac {\partial u}{\partial t} + u\frac {\partial u}{\partial x} = 0$$ the conservation law converts $u(u)_x$ to $(u^2/2)_x$. However, ...
- 13
3
votes
1
answer
111
views
What exactly is KAM stability and how can I determine if an orbit is KAM stable or not?
I have been working on the three-body problem lately and came across KAM stability. I read that KAM stability generally means that the solution is stable at different initial conditions (that of ...
- 360
1
vote
0
answers
110
views
Curvature and stability
In Topological methods in hydrodynamics 1 mentioned that "The Riemannian curvature of a manifold has a profound impact on the behavior of geodesics on it. If the Riemannian curvature of a ...
- 111
1
vote
1
answer
128
views
Calculating the Lyapunov exponents spectrum from particle trajectories
I am simulating a forced, compressible 2D flow, that is turbulent and statistically steady, but not stationary.
I want to calculate the Lyapunov exponents spectrum from the trajectories of Lagrangian ...
- 11
4
votes
3
answers
318
views
Mathematically proving that it is always possible for a rigid body to maintain its rigidity
Consider a rigid body $\mathcal{B}$ modeled by a system of $n$ point masses $B_1,B_2,\dots, B_n$ constrained to keep constant distance from each other. I wonder how it is possible to mathematically ...
- 404
0
votes
1
answer
85
views
Double Pendulum Intuition
I have arrived at the equations of motion for a double pendulum, with gravity $g$, masses $m_i$, link lengths $l_i$, angles $\theta_i$, and applied torques $\tau_i$.
Please see the diagram and ...
- 559
2
votes
0
answers
94
views
Does Poisson Distribution means the system is chaotic?
The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
- 21
3
votes
2
answers
90
views
Effect of reorthonormalisation step size when calculating Lyapunov exponents using the Gram–Schmidt reorthonormalisation (GSR) procedure
I am trying to determine the Lyapunov exponent using Gram–Schmidt reorthonormalisation (GSR), for a well-defined dynamical system (I know the differential equations etc). I believe I have implemented ...
- 31
1
vote
1
answer
118
views
Need help finding Hamiltonian for equations of motion
I have the following equation of motion:
$$\ddot \theta+\dot\theta^2\theta+k^2\theta=0.\tag 1$$
This equation is from this question. I wanted to see if I could find a Hamiltonian for this equation but ...
2
votes
3
answers
613
views
How to justify this small angle approximation $\dot{\theta}^2=0$?
Suppose the equation of motion for some oscillating system takes the following form:
$$\ddot{\theta}+\dot{\theta}^2\sin\theta+k^2\theta\cos\theta=0$$
Applying small angle approximation to $\theta$ ...
- 205
1
vote
0
answers
29
views
What are the Equations for Climate modeling of alien planet?
I am studying complex dynamic system and I would like to analyze the climating formation of a possible alien planet considering climate as a complex system .
For this I do not want to use a whole ...
2
votes
1
answer
135
views
Noether's theorem for first-order systems
In a paper I'm reading (https://arxiv.org/abs/1312.6120, equations 8-9) I've seen the following statement - given that $s$ is constant and $a,b$ are some dynamical variables obeying the following ...
