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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Is Singularity corresponds to null space and are forces causing singularity are some transformation matrix?

Some random thoughts: (Disclaimer: i am a complete noob in physics). I was studying linear transformations from linear algebra and just got this thought that there might be some set of transformation ...
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How do I solve this nonlinear ODE given the asymptotic series solutions as follows? [closed]

Differential Equation: $$-{\frac { \left( {\frac {\rm d}{{\rm d}R}}f \left( R \right) \right) ^{2}}{2\,f \left( R \right) }}+{\frac {{\rm d}^{2}}{{\rm d}{R}^{2}}}f \left( R \right) +{\frac {{\frac {...
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What is the evidence that gravitational fields don't sum up as a superposition?

Einstein's field equations are non-linear. Gravity gravitates (self-interacts). It's very complicated to solve Einstein's field equations for more than one central object. That are keystones in ...
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Self-coupling of gravity and gravitation escaping a black hole - contradiction?

The field equations are non-linear, that can be interpreted as gravity is coupling with itself, see for example here: Non-linearity and self-coupling of gravity I'm trying to understand what that ...
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Understanding EFE: RHS linear, LHS not?

Einstein's field equations are nonlinear. That means it is not allowed to add up the metric tensors. However, on the RHS of the field equations, there is only the stress-energy-momentum tensor, and it ...
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Is this version of Einstein field equations linear?

While playing around with the Einstein field equations and trying to derive the Kerr metric, I came across the following derivation from Einstein's field equations: $$R_{\mu\nu} = 8\pi \left(T_{\mu\nu}...
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How are the Bloch equations non-linear?

This question is similar to the following, but I have expanded the question moderately: Nonlinearities arising from linear equations The Bloch equations are described by the following vector equation (...
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Quadcopter motor speed mixing

I'm having trouble solving the following problem. I've designed a control system for a quadcopter, my design is built upon a nonlinear model in the affine form: $$ \dot{x}=f(x)+g(x)u $$ The system ...
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When is the motion of the perturbative term resonant with the Hamiltonian?

I am given the following Hamiltonian, $H$, which is a perturbed version of $H_0$, $$ H(\theta,I) = H_0(I) -\epsilon \cos(\theta-\Omega t)$$ where $H_0 = \frac{I^2}{2}$, $\epsilon << 1$ and $(I,\...
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How and why does a nonlinear element produce the sum and difference of two radio frequencies?

For a resonant circuit the voltages of the capacitive and inductance reactance cancel and the currents of the capacitive and inductance reactance also cancel leading to a zero reactance. Is there a ...
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Are non-linear extensions to QM equivalent to time travel?

I recently learned of the existence of Objective Collapse theories which add non-linear terms to QM to explain wave function collapse. Per the 2014 paper Treating Time Travel Quantum Mechanically, the ...
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Is there a "measure of nonlinearity" that can be measured when testing quantum mechanics?

For context, I think the comparison to tests of general relativity here is apt. There is the post-Newtonian formalism that has some well-defined parameters that can discriminate between general ...
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On the capacity for equations of motion to be contained in field equations

I've heard that the equation of geodesic motion can be derived from the vacuum Einstein field equations, although there appears to be some debate about how rigorously this can be proved, due to a ...
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Understanding the relationship of displacement and electrical field of nonlinear material using fourier

**I posted this question first in math.stackexchange but was told it might be more suitable here. I have a general formula of $$D = P_0 + \epsilon_1 E + \epsilon_2 E^2 + \epsilon_3 E^3 + ...$$ with $D$...
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Advection Term in the Lorenz 96 Model

The Lorenz 96 Model is defined as $$\frac{dx_i}{dt}=\underbrace{(x_{i+1}-x_{i-2})x_{i-1}}_{advection}-x_i+F$$ with some forcing $F$ and periodic boundary conditions so that $x_{i+N}=x_i$ for some $N$. ...
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Initial value formulation of Yang-Mills equation

In Wald Chapter 10, he discusses the initial value formalism of electromagnetism - how Maxwell's equations are actually a system of three equations plus an initial value constraint, and how we can ...
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Non-dimensionalizing laser system of diffeqs, Strogatz Nonlinear Dynamics and chaos 3.3.1 D

The system of equations in question is $$ \dot{n} = GnN - kn$$ $$\dot{N} = GnN - fN + p$$ Where ${N(t)}$ is the number of excited atoms, ${n(t)}$ is the number of photons, ${G}$ is the gain ...
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Oscillations in a Time-Varying Magnetic Field

Suppose we have a cantilever (with a magnetic moment/charge attached to one of its ends) oscillating in a magnetic field which is spatially varying in the $x$ direction and is also time dependent, ...
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Two masses connected by spring moving in a plane

Is the 2-dimensional motion of two masses connected by a spring non-linear? As far as I see it, the magnitude of the force on each mass is proportional to the spring's displacement from its ...
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2 answers
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Can the phase portrait of SHO rotate counter-clockwise? or is it the case that there can be no physical motion corresponding to that?

Framing the question In the case for Simple Harmonic Oscillation, we have the equation: $$\ddot{x}+x=0 \tag{1} \label{1}$$ (say, we put all the coefficients to be 1) Now, if we try to solve it in ...
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What is the difference between non-linear oscillator and non-uniform oscillator?

The equation for a uniform oscillator is: $\dot\theta = \omega$ which has a solution of $\theta(t) = \omega t +\theta_0$. For a non-uniform oscillator, the equation is: $\dot\theta= \omega - a$ where $...
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Solution to non-linear differential equation, non-linear oscillator

Is there any way to find an analytical solution to this equation? $$ m \ddot{x}(t) = B_0 \left( \frac{1}{x(t)^4} - \frac{1}{(L-x(t))^4} \right)$$ this is supposed to describe a magnetic oscillator ...
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1 answer
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Phase space portrait for dynamical system with Bifurcations

I have this dynamical system $$x'=y, y'=-x^3-y+mx$$ and I want to draw the phace space diagram for $m=-1/8, m=1/4,$ the bifurcation points. 1st of all I cant find what kind of bifruction I have( I go ...
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Four-wave mixing and modulation transfer spectroscopy: why do sidebands appear on the probe?

I'm trying to understand modulation transfer spectroscopy in simple terms. For those unfamiliar with it, this article gives a very good summary. To sum it up, two counterpropagating beams, a pump and ...
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Non-linear Diffusion Equation

I'm currently trying to solve the equation $$ \frac{\partial C}{\partial t}= \frac{\partial}{\partial x}\left(\frac{D}{C}\frac{\partial C}{\partial x}\right), $$ where D is a constant and $C \equiv C(...
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2 answers
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How to draw the phase plane of this equation?

Using various computational tools, it's possible to draw a phase plane from two first-order ODEs or a single second-order ODE. However, when there is a parameter in the equation and we don't know the ...
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Estimate momentum and energy in Benjamin-Ono

Consider the Cauchy problem for the Benjamin-Ono equation $$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$ where $\mathcal H$ is the ...
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6 votes
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Adjoint of a non-linear operator

I am a retired aerospace engineer, embarking on a self-study of QM. In reading S. Weinberg's book Lectures on QM (second ed.) I found the following definition (pag.65): "The adjoint $A^\dagger$ ...
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Mixing for Burgers equation in 2+1D

Let us consider the following (2+1)-dimensional Burgers-like equation: $$ u_t + (u^2)_x + (u^3)_y=0. $$ Here the unknown is a function $u= u(t,x,y):(0,\infty) \times \mathbb R^2 \to \mathbb R$. Is ...
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Scale for Hénon-Heiles Potential

I was reading about the Hénon-Heiles Potential and I read that it describes "non-linear motion of a star around a galactic center with the motion restricted to a plane" (Wikipedia Link). So ...
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What would go wrong if quantum observables were not represented by linear operators? [closed]

If quantum mechanical operators corresponding to physical observables were not hermitian, the corresponding eigenvalues may not be real. Since the eigenvalues are the outcomes of measurements of ...
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2 votes
1 answer
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A coupled nonlinear dynamical system in four dimensional phase space

I have come across a coupled nonlinear dynamical system given below $$ r\, \ddot{x} + \dot{x} = \sin y~,$$ $$ r\, \ddot{y} + \dot{y} = \sin x~,$$ where $r$ is some real number and $\dot{x}$ denotes $\...
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Why must operators in QM be linear?

Why must all operators in QM be linear (and therefore able to be represented by matrices). What is the physical reasoning behind this? Is it be possible that the non-unitary nature of quantum collapse ...
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Why is synchronisation only possible for self-sustaining oscillators

A self sustained oscillator is any oscillator which obeys the following 3 key properties (Balanov 2009): They do not damp They are capable of oscillating without being driven by an external force. ...
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How an applied magnetic field breaks the inversion symmetry in a centrosymmetric system?

I want to understand why magnetic dipole transition breaks the inversion symmetry in a centrosymmetric system and gives rise to second-order nonlinearity.
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Higher order nonlinear ultrasonic signals

With nonlinear ultrasound, the higher harmonic frequencies are used, for example, to identify defects in materials. Usually the second and third harmonic frequencies are used. If higher orders are ...
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Born-Infeld equation with a coefficient: which phenomena it describes?

Let us consider the well known Born-Infeld equation $$-{\rm div}\left(\frac{\nabla u}{\sqrt{1-\frac{1}{b^2}|\nabla u|^2}}\right) =g(u).$$ It appears quite naturally in several fields such as ...
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Is this system linear? [migrated]

$$y(t)=\begin{cases}0 \hspace{4.86cm}x(t)<0\\ x(t)+x(t-2)\hspace{2cm} x(t)\geq0\end{cases}$$ Assuming $C>0$. If $x(t) = C$ is an input, the output will be $2C$, and the output of $-x(t)=-C$ must ...
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How to find the steady state response of a Multi-Degree of Freedom (MDOF) system?

The Problem I currently have a Multi-Degree of Freedom (MDOF) system with the following equation: $$\mathbf{M\ddot{X}}+ \mathbf{D}(t)\mathbf{\dot{X}}^2 + \mathbf{C\dot{X}} + \mathbf{KX} = \mathbf{F}(t)...
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1 vote
1 answer
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Linearization of 1D maps about a fixed unstable point [closed]

Recently, I was going through the paper Controlling Chemical Chaos in a three variable autocatalator system, by Peng et al. Here are the references Although I have been introduced to 1D maps and the ...
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How do I non-dimensionalize Newton's Law of Gravitation for the 3-body problem?

I'm attempting to numerically solve the 3-body problem. Using Newton's second law, I've derived a system of 6 second order differential equations, the first three being: $$ m_1\frac{d^2x_1}{dt^2} = -G ...
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How to generate a PDE from a discrete equation in a rice-pile like model?

I am reading Noise and dynamics of self-organized critical phenomena by Albert Díaz-Guilera Here, on an extension of the rice-pile model by Bak et al demonstrating self-organized criticality. Equation ...
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Parametric Resonance Analysis using Perturbative approach

I'm reading Parametric Resonance from Landau's Mechanics Text. A similar calculation is done here. Supposing a parametric oscillator given by $$\ddot{x}(t)+\omega_0^2(1+h\cos(\gamma t))x(t)=0$$ It's ...
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What the response of unstable limit cycles look like?

Stable limit cycles generate oscillations, i was wondering what the unstable limit cycles behaviours look like? From the picture in the left, the system shows a stable limit cycle and it generates ...
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Chaotic and Ordered Random Boolean Newtorks with a fixed in-degree k and a probability p

I'm working with Random Boolean Networks, I made a python program to show the dynamics of the networks. Before coding the program I study the theory and it says that the in-degree k and the ...
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1 answer
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Higher order nonlinear stress definition

For the nonlinear case, I often find the following definition for the mechanical stress: $$ \sigma=E_2\epsilon+E_3\epsilon^2$$ The parameters $E_2$ and $E_3$ are called "elastic modulus" or &...
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4 votes
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How chaotic is the double-pendulum if the arms are not perfectly rigid?

The double pendulum is a famous example of a chaotic system. It consists of one pendulum hanging from the end of another pendulum, which in turn hangs from a fixed point. In the traditional version, ...
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Particle and metric redefine in action corresponding and beyond conformal transformation

let us beginning form action with system of gravity and scalar field. $$ S=\int d^d x \sqrt{g}(R-g^{ab}\partial_a \phi \partial_b \phi-V {(\phi)} )$$ and redefine metric $$ \tilde{g_{ab}} = f(g_{ab},\...
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3 votes
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Do these Lagrange equations of 1st kind exhibit numerical instabilities?

I followed the lead of "Theoretische Physik", 1e, 2015 by Bartelmann et al. (pp. 171 - 174) to form the set of constituting Lagrange equations of the 1st kind for the double pendulum: eight ...
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RG flow diagram plotting

I want to be able to plot a flow diagram with a given recursion relation. For example, I have the follow recursion relation: \begin{align*} \frac{dT}{d\ell} &= 2T{y_0}^2 a^2 \\ \frac{d ...
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