# Questions tagged [non-linear-systems]

The term non-linear or nonlinear has several definitions but is generally used to describe a system that cannot be approximated by a superposition principle or perturbative approach.

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### Linear Structure of Classical theory

I have been studying QFT from Timo Weigand’s lecture notes and in the chapter ‘Quantisation of spin-1 fields’, he describes the Feynman rules for QED and after some examples, there is subsection named ...
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### Do all even potentials produce periodic motion?

Consider a non-relativistic point particle of mass $m$ in 1D under the action of only conservative forces. Then by Newton's second law, the equation of motion is $$m\ddot{x}(t)=-U'(x(t)).$$ Now, do ...
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### Destruction of integrals of motion in chaotic systems: Fermi-Pasta-Ulam (FPU) paradox

I am trying to understand behavior of system studied by Fermi, Pasta and Ulam i.e. chain of oscillators interacting via nonlinear forces. I am generally not very familiar with chaos theory and ...
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### Reference for Non-Linear Water Waves

In class, my professor just mentioned that some finite-amplitude water waves were satisfied by the KdV equation. Is there some reference which shows how to derive this from $1st$ principles, and also ...
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### Studying Chaos in RLD circuit

We are currently working on non-linear dynamics (chaos theory) by analysing a series circuit including a diode (the 1N4004), a 100 ohm resistor and a 20 mH inductance. It is driven by an alternative ...
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### Minimal dynamical system with quasiperiodic oscillations

What is a minimal, explicit dynamical system (as in, a series of coupled ordinary differential equations) that exhibits quasiperiodic oscillations for some region of parameter space? Two coupled Van ...
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### The solution to the non-linear convection equation

The non-linear convection equation $$u_{t} +uu_{x}=0$$ admits implicit solutions of the form $$u=f(x-ut).$$ How does one interpret this solution intuitively? Is there an example of a solution of this ...
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### How is a quartic oscillator solved in classical mechanics?

Quantum mechanically, a quartic anharmonic oscillator with potential $$V(x)=\frac{1}{2}m\omega^2x^2+\lambda x^4$$ is dealt with perturbation theory- the approximate energies $E_n$ and energy ...
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### How do we find the equation for the gyrating motion of a particle in a uniform magnetic field and a non-uniform Electric field? [closed]

Considering the gyrating motion is not negligible and also retaining the guiding center drift, how do we get the trajectories x(t),y(t),z(t) of the particle? In this case is the variation in the ...
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### In what sense do bifurcations concern change in quality?

I've heard such vague statements several times and also read: Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family. (From ...
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### Why does a non-linear system lead to interaction and frequency mixing between input's?

When we have a system that is nonlinear and we apply a sum of two different frequency sine waves as an input, we see the output of this system has components that are at the sum frequency of the two ...
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### What is the general definition of thickness of a strange attractor?

Disclaimer: This question is cross posted on Math.SE because I don't know which site is more appropriate for this question. In Chaosbook, at page 56, it is asked to find the thickness of Rössler ...
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### What quantum phenomena violate the superposition principle in electromagnetism?

On page 11 of the 3rd edition of Electricity and Magnetism by Edward M. Purcell and David J. Morin it says: "we know of quantum phenomena in the electromagnetic field that represents a failure of ...
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### How to calculate the parameter values for which the Lorenz system is chaotic?

I was recently going via a book (Strogatz), that mentions Lorenz's attractor, and that it was found out that for values such as $a=10$, $b=\tfrac{8}{3}$, $c=21$, the system behavior is chaotic. How ...
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### Poincaré Map (Quasi-periodicity; Stability)

In a Poincaré map, when quasi-periodicity is exhibited by the dynamical system, what does it mean in terms of stability for the dynamical system?. Why is it so that as Maximum Lyapunov exponent (MLE) ...
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Do you have examples of Lagrangians/Hamiltonians used in physics with non-quadratic kinetic terms? e.g. $\dot{x}^4$ What is the origin and the interpretation of such terms?
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### Is there any nonlinear equations depending on Fourier coefficients?

A nonlinear partial differential equation is an expression depending on derivatives of $u$ $$f(x,t,u,u_x,u_t,\cdots)=0,$$ where the derivatives of $u$ can be obtained from the Taylor series of $u$. ...
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### Vacuum birefringence

Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example: ...
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### Intuition behind the meaning of Lyapunov exponents

Can anyone help me in understanding the contraction and the expansion of the phase space? what are Lyapunov exponents? and how come one understand this concept intuitively?