Questions tagged [complex-systems]
A loosely defined concept, a Complex System presents a behavior nontrivially determined by the interactions between its parts. Complex systems often exhibit emergence phenomena, such as swarming and pattern formation. In such systems, nonlinear interactions can lead to memory and feedback mechanisms, self-organized criticality, and chaotic behavior. Network theory, systems biology, and adaptive/evolutionary systems also fall under this umbrella concept.
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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:
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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 ...
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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 ...
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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 ...
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Interpretation of a time dependent and a complex-valued energy
Concerning relation 61 of Time fractional Schrödinger equation: Fox's H-functions and the effective potential,
For time dependent energies, when time tends to infinity, the energy tends to zero. ...
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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 ...
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Dimension of the eigenspace at de Almeida-Thouless calculation
I'm now reading the paper Stability of the Sherrington-Kirkpatrick solution. In the appendix, the eigenvalue problem $G \mu = \lambda \mu$ is being solved. At (A.5), the following form of the ...
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(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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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Modeling evolution of continuous mean-field model?
Suppose $f(i,t)$ indicates state $i$ at time $t$. Are there examples of exactly solvable models where $f$ is described by differential equations similar to one below?
$$\frac{\partial}{\partial t} f(i,...
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Double Pendulum Intuition
I have arrived at the equations of motion for a double pendulum, with gravity $g$, masses $m_i$, link lengths $l_i$, angles $\theta_i$, and applied torques $\tau_i$.
Please see the diagram and ...
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Does Poisson Distribution means the system is chaotic?
The Berry-Tabor Conjecture says that for classically integrable systems, the corresponding quantum systems obey the Poisson distribution for their energy-level spacing. But generally, the integrable ...
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Effect of reorthonormalisation step size when calculating Lyapunov exponents using the Gram–Schmidt reorthonormalisation (GSR) procedure
I am trying to determine the Lyapunov exponent using Gram–Schmidt reorthonormalisation (GSR), for a well-defined dynamical system (I know the differential equations etc). I believe I have implemented ...
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Is numerical optimization useful in (highly nonlocal) Lagrangian dynamics?
It is well known that many equations of physics can be formulated as an extremalization principle, i.e. that the equations arise from an action $S = \int L(t,x(t))dt$ with a time-dependent Lagrangian $...
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Systems where particles can cross a set only along certain directions
I apologize in advance for the formulation of the question, since I am mathematician, who knows little about physics.
I am curious if someone could help me identify some possible applications for ...
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Need help finding Hamiltonian for equations of motion
I have the following equation of motion:
$$\ddot \theta+\dot\theta^2\theta+k^2\theta=0.\tag 1$$
This equation is from this question. I wanted to see if I could find a Hamiltonian for this equation but ...
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How to justify this small angle approximation $\dot{\theta}^2=0$?
Suppose the equation of motion for some oscillating system takes the following form:
$$\ddot{\theta}+\dot{\theta}^2\sin\theta+k^2\theta\cos\theta=0$$
Applying small angle approximation to $\theta$ ...
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Is a damped, driven simple harmonic oscillator a limit cycle?
I've been reading about limit cycles and synchronization from Pikovsky's Synchronization in order to build a background for non-linear oscillators. What I know about the limit cycles is that they're ...
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What are the Equations for Climate modeling of alien planet?
I am studying complex dynamic system and I would like to analyze the climating formation of a possible alien planet considering climate as a complex system .
For this I do not want to use a whole ...
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Objective measure of emergent behaviour complexity
The definition of emergent behaviour I always see around is on the lines of "Emergent behaviour is behaviour that emerges only when the parts of a system interact, and such parts do not exhibit ...
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Noether's theorem for first-order systems
In a paper I'm reading (https://arxiv.org/abs/1312.6120, equations 8-9) I've seen the following statement - given that $s$ is constant and $a,b$ are some dynamical variables obeying the following ...
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Stroboscopic map
I am trying to plot the stroboscopic map of the classical kicked rotor, which is characterized by the equations:
$$p_{n+1} = p_n - \frac{dV}{dx}|_{x=x_n}$$
$$x_{n+1} = x_n +p_{n+1}$$
where $x_n$ is on ...
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Why the 3d Lorenz attractor has a butterfly shape? Why isn't it 3 dimensional too? [closed]
The Lorenz attractor has a butterfly shaped a strange attractor, but we plot it in 3D. Why is not it has a 3D shape too? It has a strange shape? It is a non-integer dimensional attractor.
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What do you call a system (like mine) that exhibits emergent physics-like behaviour from particles only described by their position? [closed]
I built a computer simulation of a 2D space which contains particles of only two types: attractive and repulsive. The particles only have an X and Y coordinate. The repulsive particles are repelled ...
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What is the simplest PDE/ODE/model I can use to understand how nonlinearities can lead to leakage of energy to higher harmonics in an oscillator?
I came across this problem in the study of surface waves in an oscillating cylindrical vessel of liquid.
There are various eigenmodes described using Bessel functions, and energy transfer can happen ...
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Do all dynamical systems have attractors?
Do all dynamical systems have attractors?
Is there any chance that there are two or more absolutely the same sets of states in one attractor?
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Distinguishing between chaos and multiperiodic oscillations from the Fourier spectrum
Consider a system which exhibits multiperiodicity, say with oscillations of the form $x(t) = \sum_{n=0} c_n \cos(n \omega_0 t)$, $\lim_{n \to \infty} c_n = 0$. The Fourier transform $\tilde{x}(\omega)$...
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Lyapunov exponent of "real life"
Today I simply forgot watching soccer WM on TV, and promptly my national team lost. Assume there is a meaningful alternative universe where I turned on the TV (quantum and relativity theorists already ...
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Literature reference: example of stable and unstable manifolds in Henon-Heiles system
There is a quite classical description of chaotic systems based on the behaviour of stable and unstable manifolds around a stationary point of the Poincaré section. It is presented, for example, [here,...
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Chaos theory: What exactly drives the future outcome?
Chaos theory states that we can't predict future because we can't measure initial conditions of a system to infinite precision. I get that.
That alone doesn't mean that the future is not determined, ...
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Prey-predator dynamical system
I'm working with a prey-predator differential equation system and I have a problem with the competitive exclusion principle.
In its simplest form, this principle states that if there are 2 predators ...
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Given a system, how to decide whether a closed orbit is homoclinic, not periodic, solely based on its phase portrait?
Background and definitions:
A system is conservative if it has at least one conserved quantity.
In a phase portrait of a nonlinear conservative system, trajectories that start and end at the same ...
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Is there any exact solution to a multivariable problem in physics not using separation of variables?
Related question (The system is not limited to integrable model, so I think this question is different)
As far as I know in quantum mechanics, exact solutions for multivariable systems (from partial ...
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Trying to prove chaotic motion from the equation of a nonlinear oscillation [closed]
So I'm given the equation of a nonlinear oscillation:
$x''+ω_0^2x=λx^3$
Assume that $x_1$ and $x_2$ are solutions to the differential equation above.
Therefore;
$x = αx_1+βx_2$
$x' = αx_1'+βx_2'$
$x'' ...
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What is difference between a monogenic system and a dynamical system?
What is difference between a monogenic system and a dynamical system?
I am confused in reading about the Hamiltonian principle because some book write system as monogenic and other dynamical.
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What is a phase space?
What is a phase space? And can the phase space be specified with x and y instead of with theta and omega?
I am currently working on a problem where I am graphing the trajectories of three masses (the ...
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Area of Phase Space and Dependence on Energy
The phase curve for a system is made for some configuration, for example - The Harmonic Oscillator. Now as we increase the energy, the phase curve enlarges i.e. area enclosed by the curve increases.
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Kasner-Arnold theorem energy role in Needham's "Visual Complex Analysis" (VCA)
In "Visual Complex Analysis" (chapter 5.X.6) Tristan Needham writes
In general, positive energy orbits in either the attractive or repulsive field $F \propto r^A$ map to attractive orbits ...
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A rule for when phase-space orbits may cross
Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (...
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What does it mean for a brain to "have more information" or be "more complex," than a volume of gas? [closed]
My big question is: are there set definitions for the concepts of "information," or "complexity?"
Before the background, here are my basic sub-questions:
Can a brain or any other ...
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Dynamics: why do physicists include derivatives like $\dot{\theta}$ in the state space for a system like a pendulum?
I come from statistics, so my experience with physics is spotty, especially on some simple stuff. I have been working on some applications related to control theory lately, and was looking at some ...
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How does the electromagnetic field behave in a complex circuit when first energized? Such as in a computer CPU booting up
From what I understand in the case of a simple circuit, the electrons in the circuit wiring are steadily excited, i.e. energized, in a fairly complex way by the external energy input, as the energy ...
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Are complexity and disorder correlated in entropy?
I am coming from the musical field, but I am looking into the topic of entropy.
In many articles from the field of physics, I keep finding what I consider a sort of misunderstanding, but I may be ...
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Is there a known method for decouple two coupled E.O.M. without making the interaction term to be 0?
I'm trying to evaluate the evolution of two scalar fields but their equations of motion are coupled via a potential term
$$ V(\phi, \psi) \supset \frac{1}{2}\lambda \phi^{2}\psi^{2}$$
From the ...
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