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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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Are all aperiodic systems chaotic?

So I understand that a chaotic system is a deterministic system, which produces aperiodic long-term behaviour and is hyper-sensitive to initial conditions. So are all aperiodic systems chaotic? Are ...
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Statistical Mechanics & Dynamical Systems

As a (theoretical) physics student I've taken (advanced) undergrad courses in both statistical mechanics and dynamical systems (which was purely mathematical, treatment of nonlinear differential ...
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Are quantum mechanical orbits specified uniquely by a Hamiltonian and initial state?

EDIT: This is completely wrong, don't bother reading. Consider a finite dimensional quantum mechanical system, say an $N$-qubit system so that $\text{dim}(\mathcal{H})=2^{N}$. Let's prepare our ...
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Is the Lorenz attractor a cyclotron?

By using a plotter to output a computer generated strange attractor solution to the Lorenz equation, that draws a line corresponding to the same fixed interval for every time step, it was found that ...
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Are there any known models with limit cycles in their RG flow?

The text-book presentation of the renormalization group (RG) leaves one with the impression that all systems will eventually flow to a fixed point. This is somewhat enforced by the phenomenological ...
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40 views

Level set of Hamiltonian are the orbits?

Just a small question : If $x(t)=(p(t),q(t))$, then the position $x(t)$ of a particle is given by $$\dot p=-H_q(x(t))\quad \text{and}\quad \dot q=H_p(x(t)).$$ In particular, if $x$ solve the previous ...
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51 views

Poincaré recurrence theorem with irrational frequencies?

The Poincaré recurrence theorem states that, for a bound phase space, the system will return to a state very close to the initial conditions, in some finite time $\tau$. For example, let's say I have ...
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25 views

Chirikov standard map derivation

This might be a stupid question, but I am having trouble understanding the derivation of Standard map by integrating Hamilton's equation of motion over one period. I am going through this dissertation ...
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30 views

If oscillatory motion is not simple (or chaotic), is it then by definition complex?

I'm trying to logically deduce or show that a specific type of motion is complex. It is two-dimensional oscillatory motion that can be expressed by coupled second order non-linear differential ...
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Euler three body problem, what exactly is it? [closed]

I have a question about the 'Euler three-body' problem. I have to write an essay about this subject for the course 'chaos theory', which is about dynamical systems and chaos. Does anybody know what ...
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33 views

Good books on Quantum Complexity and its application to Theoretical Physics?

I am about to complete my undergraduate studies and am trying to investigate some areas I might like to study in postgrad. I have read some very interesting things about the application of quantum ...
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simple models for nucleation theory

Where can I find simple dyamical models for nucleation theory. I'm particularly interested in the time dependance of the number of aggregates of different size. I've read about Smoluchowski theory ...
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38 views

Poincaré plane and Logistic Map

How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
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Poincare return map as area-preserving map

I'm trying to get some intuition into how the Poincare return map is area-preserving (when there are two momenta and two positions). Suppose $H=H(q_1,q_2,p_1,p_2)$, and let's suppose the system is ...
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1answer
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What is a quasistationary approximation

I was reading an article which states : The linear-stability analysis for this system can be performed in complete generality; but it will be best for purposes of this review to go directly to ...
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26 views

Mixing and Entropy in Dynamical Systems

I'm writting a short introductory report about chaos theory, and one of the conditions for a dynamical system to be chaotic seems to be the presence of topological mixing. Now, the document I'm using ...
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1answer
33 views

Inverse of the standard map

I'm trying to plot the homoclinic tangle that can be observed following the evolution of the unstable and stable manifolds of the standard map. The map I am using is defined as:$$ \begin{cases}p_{n+1}=...
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3answers
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Is Lyapunov function the ultimate method to assess the stability analysis of a system?

I have migrated from physics to electrical engineering and I'm seeing people in control admire Lyapunov methodology and control designs as if there is no other solutions and they consider it very sane ...
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Unpredictability, per definitions of chaotic behavior

Apparently I've been confused about the meaning(s) of "chaotic behavior". I always thought it meant that infinitesimal perturbations of a system parameter would lead to large changes in the system's ...
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Extreme values in dynamical systems

The unpredictability of chaotic systems can lead to values of physical quantities that peak up to an extremely high value for a short time. This holds also e.g. for economic dynamic systems in Terms ...
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Is *every* planar/2D system integrable?

Consider the generic following planar/2D system: $$\begin{cases} \frac{dx}{dt} = A(x,y)\\ \\ \frac{dy}{dt} = B(x,y), \end{cases}$$ where $A,B$ are two functions. Reading Classical Mechanics by ...
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33 views

Kuramoto model limit $N\rightarrow\infty$

Is it possible to say anything about the behaviour of the Kuramoto model for large but finite $N$ based on an analysis of the model obtained in the limit $N\rightarrow\infty$?
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Susceptibility in Bifurcations

In dynamical systems a bifurcation describes the phenomenon where, as a system parameter is varies, a qualitative property of the system might be altered. Similarly, in statistical physics, a ...
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2answers
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Can phase transitions occur in open systems?

Is it possible that in open systems can occur phase transitions if the required conditions (i.e., temperature and pressure) are met? Are there examples?
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How would I calculate the maximum complexity of living system based on entropy increase acceleration? [closed]

Living systems locally decrease their own entropy by increasing the entropy of their environment by larger margin, creating a local minimum in entropy. The higher the complexity of the living system, ...
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A question over Liouville’s Theorem

I have some doubts about Liouville theorem, probably its just something conceptual. So: I know that for a system in which Liouville’s theorem holds, the volume in the phase space is conserved. But ...
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Life around a different element from Carbon [closed]

On Earth, life developed around Carbon (and Hydrogen and Oxygen). I guess this depends on the availability of those elements, but also on the spectrum of radiation and Temperature and Pressure. Under ...
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1answer
41 views

Why is dynamics first order in phase space?

I have watched some lectures in which the lecturer said that system dynamics are (generally?) first order in phase space, forming a system of coupled differential equations. At a basic level I see ...
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1answer
32 views

Contradiction? Synchronized (steady) state and power flow

I am trying to investigate power flow in a simplified power system model with a few connected rotating motors. The dynamics is captured in the differential equations $$\frac{d^2 \phi_i}{dt^2} = ...
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1answer
46 views

Defining a 'small disturbance which dampens in time' while identifying stable points in a nonlinear system

I'm reading the book "Nonlinear dynamics and Chaos" by S Strogatz. In section 2.2, titled "Fixed points and stability", he defines equilibrium points as solutions where ...all sufficiently small ...
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3answers
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Physical intuition behind Poincaré–Bendixson theorem

The Poincaré–Bendixson theorem states that: In continuous systems, chaotic behaviour can only arise in systems that have 3 or more dimensions. What is the best way to understand this criteria ...
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139 views

When is the ergodic hypothesis reasonable?

Consider an Hamiltonian system. In which circumstances is it possible to assume that all the states belonging to the hypersurface $H=E_0$ are equally visited? Is it necessary to have a very high ...
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3answers
164 views

Hyperbolic harmonic oscillator

The classical harmonic oscillator can be associated to the differential equation: $$y''+\omega^2y=0$$ and solutions $$y=A\cos(\omega t)+B\sin(\omega t)$$ or $$y=A\cos(\omega t+\delta)$$ The harmonic ...
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1answer
50 views

Physical significance of orbital stability

I saw the orbital stability in Wiki, I just understand it from mathematics angle. But in physical, what is its mean? Since I saw many paper talk about the stability of Schrödinger equation, I think ...
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Wigner 's unreasonable effectiveness of mathematics in natural sciences [closed]

This question is related to Wigner's problem, related to the unreasonable effectiveness of mathematics in natural sciences. Understanding a phenomenon means constructing a mathematical model , and ...
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Usage of Floquet's Method

I'm treating with a nonlinear system of ODE, in which one of my fixed points is non-hyperbolic, that is, its eigenvalues has ($\Re(\lambda_{1,2}) = 0$). Therefore, I cannot say anything about its ...
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2answers
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Closed gravitational orbits and gradient systems

I am currently studying non-linear dynamics on my own time. One of the theorems in the material is that systems that can be written as gradient problems cannot have closed orbits i.e. systems like $$\...
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1answer
54 views

Why do my calculation of Lyapunov exponents strongly depend on the number of iterations?

I have a project in my school so I have to calculate my arranged double pendulum system's Lyapunov exponents, I refer to this method. http://sprott.physics.wisc.edu/chaos/lyapexp.htm As the title ...
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1answer
62 views

Control systems from a physicists perspective

I am highly interested in the study of control systems theory. However it seems that almost all books are written by electronics or mechanical engineers. Due to this they generally omit many things. ...
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Difference between unstable fixed point and chaotic point

I am reading the Scholarpedia article on Lyapunov exponents: Given a dynamical system $$ \dot{\vec{x}}=\vec{F}(\vec{x}) $$ and a fixed point $\vec{x}_0$ such that $\vec{F}(\vec{x}_0)=\vec{0}$, the ...
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What does it mean by 'unfolding of a pitchfork bifurcation'?

I am analyzing a dynamical system, where due to a small imperfection the original perfect bifurcation structure gets disturbed and leads to complex emerging bifurcations, one of which is a ...
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Existence of Black hole radiation (through conservation) and Analysis of classification of black holes in classical dynamical system

I was taking nonlinear dynamical system where I found a confusion with black hole. (The dependency here was with respect to single $t$ and space was "flat", in a sense that the metric does not carry ...
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1answer
139 views

Ott-Antonsen-Ansatz

im reading a paper https://doi.org/10.1063/1.2930766 about the Ott-Antonsen-Ansatz that is used to describe the dynamics of global coupled oscillator. There is a computational step from equation (4),(...
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1answer
156 views

Is there a relation between large-scale oscillations and small-scale oscillations?

From Neural oscillation - Wikipedia: Oscillatory activity in the brain is widely observed at different levels of organization and is thought to play a key role in processing neural information. In ...
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1answer
22 views

Multi-stability potential is suitable for interdisciplinary branches of science?

The potential function $V(x) = c(x-a)^2(x-b)^2$ with constants $a,b,c$ and some variable $x$ has two minima: One at $x=a$ and another at $x=b$. When plugging this potential function in Newton's ...
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1answer
59 views

Dimension of Poincaré Map

I am used to seeing bi-dimensional Poincaré maps, as the ones shown in this post: Poincaré maps and interpretation In that example, one manages to draw a bi-dimensional map because the number ...
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1answer
50 views

Question about a system with all bounded orbits closed and maximal integrable

Given Hamiltonian system with $2n$-dim phase space, if there exist $k\ge n$ independent integrals of motions then we call it integrable Hamiltonian system. The largest number of independent integrals ...
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1answer
69 views

Classification of fixed points in 4D phase space

The usual classification of fixed points as used in linear stability analysis is based on planar systems (un-/stable node, un-/stable spiral point, saddle). I need to extend this classification to a ...
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1answer
52 views

Book on Non-Linear dynamics [duplicate]

I am currently planning on self studying Non-linear dynamics, with an intention of developing a decent idea about it and see how its applied. I don't want to be too rigorous in my dealing with the ...
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1answer
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Can a first-order autonomous system not at a fixed point, transition to a fixed point?

The following is from Introduction to Dynamics, by Percival and Richards: At each zero $x_{k}$ of the velocity field $v\left[x\right],$ $$v\left[x_{k}\right]=0,$$ so that a system ...