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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Linear system in polar coordinates [closed]

Unlike the Cartesian coordinates, I find navigating through polar coordinates difficult. Is the system defined by the following Lagrangian $L$ defined in polar coordinates linear? $$L = \frac{1}{2} m \...
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Is linear response only the first term in a series expansion?

So in the theory of linear response, the goal is to look at how certain dynamical variables (or operators in QFT) respond to an external source. To be more concrete, suppose that $x$ obeys (in index ...
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Fréchet derivative of a operator $E: H_{per}^{1}\left([0,L]\right) \longrightarrow \mathbb{R}$ [migrated]

Define the operator $E: H_{per}^{1}\left([0,L]\right) \longrightarrow \mathbb{R}$, given by $$E(u)=\frac{1}{2}\int_{0}^{L}(u_t^2+u_x^2+\frac{1}{2}(1-u^2)^2)dx,\; \forall \; u \in H_{per}^{1}\left([0,...
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Is it possible to find the linearized operator through the conserved quantities?

Let $$u_{tt}-u_{xx}= u-u^3 ,\: (t,x)\in \mathbb{R}\times \mathbb{R}.$$ I know that the linearized operator around a solution $u$ is given by $$\mathcal{L}=\frac{\partial^2}{\partial t^2}-\frac{\...
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Doubt on Lax formulation of Korteweg–de Vries equation

The Korteweg–de Vries equation is given by: $$\frac{\partial u(x,t)}{\partial t}-6u\frac{\partial u(x,t)}{\partial x}+\frac{\partial^3 u(x,t)}{\partial x^3}=0$$ This equation can be formulated using ...
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How can I use bifurcation analysis of the Lorenz system in calculating the fractal dimension by the Spectral decay coefficient method?

Discrete Fourier transform represents data by a superposition of sines and cosines that have various amplitudes and frequencies. With time series of length N, the range of frequencies that can be ...
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How to express the “size” of measurement noise in a dynamical system

I have a discrete [non-linear] dynamical system $x_{n+1} = f(x_{n})$. There is measurement error, so my observables are a time series $\left\{ \hat{x}_{n}\right\} _{n=1}^{N}$ where $\hat{x}_{n}=x_{n}+\...
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Resistive forces on Simple Harmonic motion

How is a simple harmonic motion affected by resistive forces? In this case, a spring block system is placed on rough horizontal surface. How to derive the block's displacement equation? I couldn't ...
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Are all familiar symmetry transformations, when they act on fields, linear? [duplicate]

Consider symmetry transformation acting on a field or a set of fields. For example, a gauge transformation of the form $$\phi^\prime_a(x)=U_{ab}(x)\phi_b(x)$$ where $U(x)$ is a matrix with elements ...
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Periodic traveling waves of the form $\phi(x,t)=\psi_c(x-ct)$ for a $\phi^4$ model

Consider \begin{equation}\label{1} \partial^2_t\phi-\partial^2_x\phi=\phi -\phi^3,\: \ (x,t) \in \mathbb{R}\times \mathbb{R} \hspace{30pt}(1) \end{equation} the $\phi^4$ model. I know that $$H(x)=\...
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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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109 views

Phase diagram method

I was trying to find the famous attractor solution of the inflaton field which follows the equation $$\frac{d\dot{\phi}}{d\phi}=-\frac{\sqrt{12\pi}(\dot{\phi}^2+m^2\phi^2)^{1/2}\dot{\phi}+m^2\phi}{\...
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Why would we want to calculate the Lyapunov exponent for experimental data?

Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even ...
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How are jerk equations connected to chaos theory?

I read in this Wikipedia article: It has been shown that a jerk equation, which is equivalent to a system of three first-order, ordinary non-linear differential equations, is the minimal setting ...
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3answers
112 views

What are the necessary and sufficient conditions for a motion to be periodic?

Consider the following idealized motions (i) The motion of a bob attached to a spring on a horizontal frictionless table, (ii) the motion of a pendulum with an arbitrary amplitude without air ...
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Physical meaning of third derivative with respect to position

I currently on a numerical solver for the KdV equation which reads $$ u_t + uu_x = u_{xxx} $$ I was wondering the physical sense of this third derivative with respect to $x$. I know that the $uu_x$ ...
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Instability of coupled non-linear oscillators

Consider a bunch of interacting oscillators (e.g., a chain of atoms), interacting due to anharmonicity in the potential energy. You can Taylor expand the force on each oscillator about equilibrium ...
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Is the quantum dynamics of a system of interacting particles linear or non-linear?

As I understand it, the linearity of quantum mechanics is considered to be an inviolable principle - e.g., this paper - because (among other things) causality would be violated or and/or superluminal ...
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What can one conclude about the stability of limit cycles without the use of numerical methods?

Let's assume one asserts the existence of a closed orbit by applyling the Poincaré-Bendixson theorem to a trapping region $R$ that is constructed such that all phase vectors on its boundary point ...
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Lorentz Equation Symmetry

I was going via Lorentz equation & learning the topic on Symmetry, what I couldn't understand is how did they performed this type of substitution & what is the philosophy behind this way of ...
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What is the strain energy around a stressed configuration ? Is $W_{13}=W_{12}+W_{23}$?

Strain elastic energies are generally defined around a stress-free configuration. Is it possible in some case to define it around a stressed configuration ? Meaning in general we have $W=W(\mathbf{F}...
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Calculating the absorption cross-section for Helium in a strong oscillating external IR field -> AC Stark Shift (Autler-Townes Splitting)

I try to obtain the absorption cross-section for atomic helium, in a strong and oscillating IR field, for when a second - XUV - pulse is probing the system. I think the only way of doing this is to ...
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Gravitational non-linearities and Dark Matter/Energy

I had read Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies (https://arxiv.org/abs/1909.00095), was wondering Is there good reason to think that gravitational non-...
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Is there anything more chaotic than fluid turbulence?

Fluid turbulence is a highly complex and non-linear chaotic phenomenon. Great difficulties and complications are encountered when trying to accurately and robustly calculate or simulate fluid flows, ...
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51 views

Could different outcomes have different physics in Wigner's friend?

Could different outcomes have different physics in Wigner's friend? Physicist Eugene Wigner said that consciousness was fundamental for physics and that laws of physics existed because of it. He said ...
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Confusion about non-ohmic conductors

Could someone please explain what happens in a non-ohmic conductor when the voltage is dropped in terms of current and resistance? It would help me a lot if it were done in detail. I don't understand ...
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Intuition behind focusing vs defocusing in integrable systems like NLS, KdV, mKdV

The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description) Non-Linear Schrodinger equation Korteweg-de Vries equation ...
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Natural phenomena with cubic behaviour [closed]

I'd like to know which natural phenomena (in planet earth) may be described with a cubic function/polynomial? or is there not any. Accelerated movement is quadratic. Work, is also quadratic. the ...
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Dimension of the constant in Born-Infeld nonlinear electrodynamics

As I know, based on the Lagrangian of Born-Infeld electrodynamics, its constant which shows the strength of electromagnetic field should have the dimension of inverse of length, but in some papers I ...
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How to get the principle stretches of compressible Neo-Hookean material under uniaxial extension?

As the title described, how the principal stretches of a compressible Neo-Hookean material undergoing uniaxial extension are derived from the constitutive model as below? $$ \lambda_1 = \lambda; \...
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34 views

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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70 views

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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291 views

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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Non-quadratic kinetic energy [closed]

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?
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Distinguishing a LTI from not with unknown inputs

Linear time invariant (LTI) systems are a staple of physics. They appear in many situations. But how do you know a system is a LTI? In particular, if you are provided with a black box which ...
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Infinite series vs compact representation

I understand the attractiveness and usefulness of infinite-series expansions such as Taylor expansions, but I wonder if they sometimes hide important aspects of the described system. For example, ...

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