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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From phase portrait to equation
Given a certain phase portrait/phase space, what is the right approach in order to find an equation $\dot{x}=f(x)$ (or a set of equations $\dot{x_n}$) with a flow consistent with that portrait?
More ...
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Why are solitons reluctant to disperse their energy?
Can somebody explain in words alone why solitons survive in water so long? Are they moving with low friction through the surrounding water imparting little energy to it?
user547468
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Is the collapse of the wavefunction really part of the quantum theory?
The collapse of the wavefunction by comparing it with the Schrodinger equations has some differences: it is higly non-linear while the Schrodinger equation is linear, it is non-local as proven by Bell'...
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Well-posedness of initial value problem in chaotic systems?
Quoting Wald from his seminal textbook on general relativity (Chapter 10):
First, in an appropriate sense, "small changes" in initial data should produce only correspondingly "small ...
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Can physical waves develop an infinitesimally narrow singularity?
consider a wave described by some field. the mod square of this field often corresponds to energy density or particles density. Mathematically, the total energy or particle number could be finite even ...
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Charge in non-linear Electrodynamics and GR
How do we define charge in a theory of curvature with non-linear electrodynamics?
For example, suppose we have an action
\begin{equation}
S = \int d^D x \sqrt{-g}\Big( R - k\Lambda + (cF)^n\Big)
\end{...
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A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - part 2
A brick sliding in an horizontal plane after an initial push (under Coulomb's dry friction) - part 2
Intro
This is a follow up of a previous question.
Main Body
From these Wikipedia sites: Contact ...
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Applications of systems exhibiting strange attractors
I have always thought of strange attractors as mathematically interesting and aesthetically pleasing phenomena. In investigating a system for my PhD research, I have surprisingly stumbled across the ...
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Normal coordinates in context of gravity
When doing coupled oscillator problems. Normal coordinates are those such that couplings (the off-diagonal terms in the matrix) appear vanish. This is diagonalization of the eigenvalue problem.
In ...
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Term similar to potential energy but for oscillation cycles instead
I'm not sure if this question makes sense, but similar to how one can assign a potential energy based on the instantaneous spatial configuration of a system, which gives us insight into what state the ...
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Attractor in Poincaré section
I have a dynamical system which is shown by 2 second order differential equation which are coupled:
$$\ddot{x} + \gamma\dot{x} + \frac{1}{m} \frac{\partial H}{\partial x} = 0$$
$$\ddot{y}+ \gamma\dot{...
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Demonstrate hysteresis of Duffing equation in numerical solution
Objective
I would like to model the following Duffing equation using Runge-Kutta 4 algorithm :
$$
\ddot{x} + 2\mu\dot{x} + \gamma\dot{x}^3 + \omega_0^2x + \alpha x^3 = k\cos{\omega t}
$$
I am using an ...
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Nonlinear dynamics: analytical solutions to a sinusoidally forced
I asked this question on math stack exchange but I wanted to repeat it here, since I was studying a physical system when I came across the following differential equation:
$$ \ddot{\theta}+\alpha \dot{...
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Derivation of the Fokker-Planck equation
A nonlinear dynamical system is considered
$$ dx/dt = f(x) + z(x)p(t), $$
where $p(t)$ Gaussian noise with zero mean and exponential correlation function.
How I can derivation of the Fokker-Plank ...
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How can I estimate the velocity of a conveyor belt relative to a laser line profiler?
I have a laser line profiler scanning a conveyor belt:
At every time, step i: $t_i = i \Delta t$ the line profiler measures n points on the line:
$$
(Y_j,Z_j,I_j)
$$
Where $Y_j$ is the position along ...
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Find closed orbit problem (Strogatz 8.3.2) [closed]
I'm having trouble solving this excercise from Strogatz
Consider the following system for a chemical oscilator:
$$
\dot x= a -x+x^2y
$$
$$
\dot y=b -x^2y
$$
Where $a,b>0$ are parameters and $x,y\...
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About barium titanate
Is barium titanate (BaTiO3) thin film martensite? Recently, Everhardt et al., in PRL (123, 087603 (2019)), show that as temperature increases, the domain evolution shows period-doubling bifurcation ...
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Non-linear crystals' interaction with light
This is really just a general question because we've been seeing non-linear crystals in a crystallography class, very briefly.
I was wondering how can we possibly understand the unique way non-linear ...
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Why do people say the dynamics of quantum mechanics is always linear?
This statement seems false. An example of a non-linear equation governing the dynamics of a quantum system is the Gross-Pitaevskii equation.
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How to understand the non-linear calculation model of demagnetization curve correctly?
in the doc of MotorCAD, I found that the demagnetization curve can be calculated by like this non linear model:
In above formula, the Br at specific temperature can be calculated like this:
And the ...
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How do gravitational fields combine together in GR?
When we have 2 massive bodies coming close together say 2 black holes or 2 massive stars, how do their respective metrics/spacetime curvature combine in the space in between them?
Do we write
$$G_{\mu\...
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Understanding linearity of Maxwell's equation compared to non-linarity of GR
In this post, it is mentioned that a linear equation means that the solutions 'do not interact with each other' or 'do not know' about each other. But we know that Maxwell's equations are linear ...
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Non-linear time and concurrent perceptions of reality [closed]
I am asking about what the fact that a photograph and a physical space can exist, in what we perceive to be different moments in linear time (with the photo being made from what we regard as our ...
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Can spring constant change by twisting or unwinding spring?
I am trying to study analytically the behaviour twisting springs and I noticed that if I consider mass and shape of spring the winding and unwinding of spring affects it's mass distribution and was ...
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How are far from equilibrium systems studied analytically?
I've read about stuff having to do with complex systems where some pretty wacky stuff happens, mostly involving "phase changes", which as I understand don't really have much to do with ...
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How would you interpret the quantum limit cycle represented by the Wigner function?
This paper by Arosh et. al. discusses the emergence of limit cycles in the quantum phase space (the Wigner function) for nonlinear oscillators.
(The quantum limit cycle of the quantum RvdP oscillator ...
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Nonlinear physics [closed]
Hey I wanna start studying nonlinear physics, and to be honest I don't know from where to start, I need books for beginners that explains things in general about the nonlinear science branches, so ...
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Symmetry and integrability in classical Hamiltonian
I am trying to understand the behaviour of an Hamiltonian system I'm simulating. I will give a quick context setting. The system is defined as
$$
\mathcal{H}(\mathbf{z};\mathbf{z}^*) = \sum_{i=1}^{M}...
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How to linearise on Lagrangian level?
Consider a Lagrangian density
$$\mathcal{L}(\phi, \nabla \phi) = \frac{1}{2} \, g^{\mu \nu} \, \partial_{\mu} \phi \; \partial_{\nu} \phi + V(\phi) \tag{1}$$
The equation of motion (EOM), i.e. the ...
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What are some good resources to learn fluid mechanics? [duplicate]
I know that there are a lot of resources out there to be explored and I have gone through several of them. What I want is some resource where fluid mechanics is treated,
from a geometric viewpoint,
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The "small amplitude" assumption in the derivation of the wave equation for the string
I am reading about the wave equation for transverse waves in a string from the book Mathematics of wave propagation (2000) by J. Davis. On page 10, just before the derivation of the (one-dimensional) ...
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Are there Dirac equations for different energy-momentum dispersion relations?
When I was introduced to the Dirac equation they wrote a PDE such that plane waves satisfy $E^2 = P^2 + m^2$. They went on to show that other options (ie Klein–Gordon) don't have spin.
Are there Dirac ...
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How to determine if gravity is roughly linear?
The Einstein field equations are famously nonlinear, which is one of the properties that makes them difficult to solve. I know (or at least I believe) that a linear system's behavior is roughly ...
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Is there a second-order non-linear addition to Maxwell's equations?
Maxwell's equations are famously linear and are the classical limit of QED. The thing is QED even without charged particles is pretty non-linear with photon-photon interaction terms. Can these photon-...
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Do all nonlinear systems store energy?
I would like to clarify, this question comes from my own curiosity while solving for nonlinear differential equations. I have noticed that I lack the fundamental understanding of linearity/...
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Resources on Phase Ordering Dynamics and Non-Linear System
I am doing a course on Non-Equilibrium Physics. Prof. was initially following Strogatz but has now started teaching Phase ordering dynamics, Cahn-Hillard equation and all?
I can't seem to find a good ...
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Burgers' equations and shock waves
Given Burgers' equation, $m_{\tau} + mm_x = 0,$ one expects to have discontinuities and thus shock waves in the case the initial conditions are smooth. For example, one may take $m_0(x) = \sin(x), x\...
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Burger equation and shock waves
Given the burger equation, $$m_{\tau} + mm_x = 0,$$ one expects to have discontinuities and thus shock waves in the case the initial conditions are smooth. For example, one may take $$m_0(x) = \sin(x),...
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Why are the equations of motion for a free quantum field theory always linear?
So far all the Lagrangians I have come across in my studying of quantum field theory have had a free theory whose equations of motion are linear. A linear free theory is of course desirable from a ...
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Why do shockwaves refract when they travel into the ground?
If a shockwave from something like an explosion travels into the ground, why will it refract? The speed of sound is far different in the ground, but what would make it refract? I can’t seem to find ...
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How are shockwaves able to refract?
How are shockwaves able to refract? As said here,
When two shock waves collide, they interact with each other and produce complex patterns of compression, rarefaction, and reflection. The resulting ...
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Dispersion relation for non-harmonic waves
This question is related to my previous one.
The entire linear theory of waves is built on dispersion relations, which represent the algebraic dependence of frequency on wave number. That is we ...
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How do shockwaves interact?
As seen here, there are two T-38's going supersonic. What happens when those shockwaves interact? They seem to dissipate in some places on this photo when they interact. Any source online says that ...
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Validity of approach to nonlinear, driven, damped oscillation amplitudes in L&L
In §29 of L&L mechanics, the authors discuss an approach to estimate the resonance amplitude of the equation
$\ddot{x}+2\lambda\dot{x}+\omega_0^2x = \frac{f}{m}\cos(\gamma t)-\alpha x^2-\beta x^3$ ...
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Entropy in chaos dynamics
I'm curious about how entropy is defined within chaos theory. Are there analogous laws similar to the second law of thermodynamics? How do we define steady-state or equilibrium within the state space ...
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Does every shockwave have an expansion wave behind it?
Do all shockwaves have an expansion fan or expansion wave behind them? Does the air always expand behind a shockwave?
I assume that the strength of the expansion wave depends on the strength of the ...
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Can protrusions on the smooth surface of a floating (or flying) body not slow it down, but accelerate it?
These protrusions are sure to create turbulent vortices. But what if these additional vortices can somehow lead to acceleration?
Additional clarification
It is clear that moving protrusions such as ...
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Self-similar solution of the second kind
I have a problem trying to understand the procedure for using self-similar solution of the second kind. More specifically, I was reading about an equation of this form,
$$\partial_t{d} + \frac{1}{r} \...
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Applications of Schrodinger's to dark solitons [closed]
The Schrodinger equation (SE) admits dark solitons as particular solutions. The SE and the The Korteweg-de Vries (KdV) equations can be used to model them.
Questions:
What are the applications of ...
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How to find the stability of time dependent Lyapunov equation?
After linearization of the nonlinear equations, I want to find the covariance matrix $v$ through the numerical solution of time dependent Lyapunov equation, $$dv/dt=a*v + v*a'+ d,$$ where $a$ is my ...
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