# 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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### 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, ...
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### 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-...
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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 ...
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### 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 $L$, hence in an introductory paragraph explains why, when studying stability of a ...
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1 vote
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### 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 ...
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### 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 ...
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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 ...
1 vote
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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 ...
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'' ...