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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18 views

No even-order harmonics in alkali gases? (Intuition for “symmetry in time domain'')

The polarizability can be expanded in the from: In Alkali gases, it is said that $\chi^3$ can be nonzero, and the combination of multiple waves can produce four-wave-mixing. But what about $\chi^2$? ...
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Does QED evolve unitarily after the Schwinger limit?

If QED becomes nonlinear after the Schwinger limit, shouldn't QED no longer be unitary (above the limit) since linearity is a requirement of a unitary operator (and vice versa)? Does this mean that ...
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Solutions of non-linear equations in QFT and interpretation of particles

One of the simplest QFT model is $\lambda \phi^4$ theory. $$ S = \int dt d^dx\; \left(\frac{1}{2}(\partial_\mu \phi)^2 - \frac{1}{4!}\lambda \phi^4 \right) $$ Equation of motion: $$ \Box \phi = -\frac{...
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Mean Field Phase Transition of Spin Systems on a irregular graph

In the Ising model w/o external field, if we use mean-field approximation, and have the regular graph lattice... then we can use the symmetry argument to recover the fact that $$\bar{x} = \tanh{(\beta ...
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Schrieffer-Wolff transformation for interacting bosons on a lattice

I have the following Hamiltonian for bosons in a lattice model with on-site interactions $$H = H_0 + gV,$$ $$H_0=\sum_{n=1}^3 \left[ \omega_0 a_n^\dagger a_n + \Omega \right( a_n^\dagger a_{n+1} + a_{...
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Can a system of cubic anharmonic oscillators have multiple stable equilibria?

Consider the Hamiltonian for a System of $N$ anharmonic oscillators $H= \sum_{i = 1}^N (\frac{p_i^2}{2m_i}+\frac{1}{2}k_iq_i^2)+\sum_{i,j=1}^N b_{ijk}q_iq_jq_k$ with specific constants $k_i,b_{ijk}, ...
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What does it mean if a circuit has 2 resonances?

I was going through capacitive coupling of qubits, and someone said to me that if we couple qubits with a capacitor we have 2 resonances. I do not understand, what does it mean to have 2 resonances.
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Kuramoto model: a question about synchronization and critical couplings

Conisder a Kuramoto model on a graph with adjacency matrix $A$, $$\dot{\phi}_i=\omega_i+\lambda\sum_jA_{ij}\sin(\phi_j-\phi_i)$$ Consider the following order parameter, $$r(t)=\frac{1}{N}\sum_je^{i\...
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Applications or work that has been done in Non-Linear dynamics or Chaos

I have almost finished my Non-Linear Dynamics course. I'm really interested in working on this field but first I want to see and study some of the work or application that has already been done in ...
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Electric field of a capacitor with a non-linear dielectric

Let's consider a parallel plate capacitor with a non-linear dielectric: The electric field between the plates will be equal to: $$E = V/d$$ where d is the distance and V the voltage between the ...
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Dynamical system fixed point by perturbation?

Suppose I have an non-linear autonomous system : $$ \dot{x}_i(t) = f(x_i(t)) + \lambda g(x_i(t)) $$ I am interested in finding its fixed points. I want to know if the following method of ...
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When can complex real world phenomena be modeled as simple low dimensional systems?

My main interests are biological systems, but the question is general. I was trained in computational biology, and virtually all quantitative models of biological processes I've encountered in my ...
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Why is Newton's second law with potentials not a linear equations?

I was trying to learn Quantum physics by myself using MIT's 8.04 course. I came accross this equation: I don't understand why the above is true. I understand the definition of linearity. But I don't ...
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Ampere's Circuital law in non-linear and dispersive medium

For a linear and non-dispersive medium $\vec{B} = \mu \vec{H}$. So amperes circuital law in integral form (without Maxwell's correction term) can be written in two ways $\oint_C \vec{B}\cdot\vec{dl} =...
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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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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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70 views

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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112 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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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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77 views

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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151 views

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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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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36 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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78 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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411 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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