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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316 views

Why does the non-linearity of the string action prohibit stretching due to strong excitations?

From 't Hooft's String Theory lecture notes on page 8 (paraphrased): To understand hadronic particles as excited states of strings, we have to study the dynamical properties of these strings, and ...
5
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1answer
74 views

Pink noise in low-dimensional systems

Pink noise (1/f) is often cited as a signature of complex or critical systems. Is it possible for a low-dimensional time-independent first-order system to generate pink noise? Intuitively it seems ...
3
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1answer
61 views

Wave vector relation in nonlinear material

A light wave ($k_1,\omega_1$) travels in a medium of refractive index $n_1$ and then encounters a nonlinear medium ($n_2$) under the angle $\theta_1$. Snell's law tells us the wave's direction in the ...
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1answer
128 views

How to simulate pendulum movement with high amplitude

I need to make a C# simulator for a simple pendulum. I have been searching the web for 3 days and I am stuck. The problem is I have found many equations that would give the angle position as a ...
1
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1answer
41 views

Tangent and Normal accelerations position estimation

How can I derive a particle position given it's last known positions (x,y), velocities in it's components (vx, vy), tangential and centripetal (normal) accelerations? (this is the only available data) ...
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1answer
256 views

Does Lyapunov exponent equate to exponential inflation?

Physics can be modeled by dynamical systems $f^t(x)$ as well as by PDEs. The most common dynamical system has hyperbolic fixed point and can be an attractor or a repellor. The dynamics at repellors ...
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1answer
70 views

What is a stroboscopic map?

I have an assignment where I'm supposed to generate a "stroboscopic map" of some orbits of a dynamical system. I have a hard time finding information about exactly what this kind of map is on the ...
0
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1answer
86 views

Time it takes for a mass in a linked pendulum to flip?

I have created Mathematica code that simulates a double pendulum. So I've numerically solved for $\theta_{1}(t)$ and $\theta_{2}(t)$. I have also found the momentum from the Lagrangian as well. My ...
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1answer
30 views

Studying dynamic elasticity for finite deformations

this is not a question asking for help with a problem but one asking for help where to begin serious study of elasticity, particularly that applied to dynamic systems. Most textbooks about elasticity ...
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156 views

Driven Pendulum

If the point of suspension of a pendulum is driven periodically in the vertical direction , we can derive the equation of motion for the suspended mass to be of the form, $\ddot{\theta}(t) + (a-b\cos{...
4
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0answers
53 views

Solutions of nonlinear systems invariant wrt. perturbations (looking for applications)

I want to ask if the following purely mathematical problem (that I'm working on) might have some applications to physics. The problem in a nutshell: describe properties of solution sets of real ...
3
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0answers
96 views

Non-linear Wave Equation - Numerical Methods

Motivation: I'm working with a highly non-linear spherical wave-like equation (second order PDE). The equation can be written on the form $$\ddot{u} = f(t, \dot{u},\dot{u}',u',u'')$$ where $'=\frac{...
3
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0answers
63 views

Reference for the Landau-Lifshitz system

I'm interested in understanding the dynamics of the discrete Landau-Lifshitz system. It's solutions to equations like $$\frac{\partial X_n}{\partial t} = X_n\times (X_{n-1}+X_{n+1})$$ where the $X_n$ ...
2
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0answers
17 views

Fermi-Pasta-Ulam for the beam equation

The Fermi-Pasta-Ulam numerical experiment is based upon the discrete wave equation, with a small non-linearity added to the forcing term. Does anybody know of similar research performed on the beam ...
2
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0answers
29 views

Why do vortices scatter at right-angles

I have been taking a course on non-perturbative physics and currently the teacher is away so I cannot ask him. In the lectures, he made the claim that a pair of vortices in the abelian-Higgs model ...
2
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0answers
43 views

integrability and area-preservation property of dynamical systems

Suppose we have a map defined on a plane, $x_{1}=f(x_{0})$, where $x \in \mathbb{R}^{2}$. Assume it is integable: there exists a function $I$ of the phase space variable $x$ such that $I(x)=I(f(x))$. ...
2
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0answers
142 views

What is the mechanism of subharmonic oscillations?

It's clear to me from linear systems theory that energy manifested within a fundamental mode of resonance can saturate with the excess energy spilling over into harmonic frequencies greater than the ...
2
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0answers
161 views

What are the equations of motion that model near light speed orbits of a massive body about incredibly massive bodies?

In Kip Thorne's recently published book, The Science of Interstellar, he describes, by means of an illustration, the complex nature of a spacecraft orbiting a massive black hole with velocities ~ 0....
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0answers
48 views

Categorization of electromagnetic solitons?

I've seen over the years several mentions of electromagnetic solitons that appear in the high-intensity regime (where vacuum polarization becomes important). Some of these are coupled with plasmas, ...
2
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0answers
57 views

Why can any pair of master coordinates be used to calculate a nonlinear mode of a nonlinear dynamical system?

This is a question I have been asking myself for some time since the following technique is often used in the nonlinear dynamics community, but never managed to get an answer why it could be applied. ...
2
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0answers
194 views

Trying to solve 2D Toda Lattice Equation with Lax Pair Approach

I am working on this Hamiltonian: $$ H = \frac{p_1^2 + p_2^2}{2m} + e^{q_2-q_1} + e^{q_2} + e^{-q_1} -3 $$ Thank you for the hint that it is a modification of the Toda Lattice Equation. Let me sketch ...
2
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0answers
107 views

KdV equation and classical linear wave equation

Like we know, the standard form of KdV equation is $$u_{t}-6uu_{x}+u_{xxx}=0,\tag{1}$$ where this equation describes a solitary wave propagation and $u=u(x,t)$. On the other hand, we know the ...
2
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0answers
107 views

value of $\omega$ in nonlinear equation

In this article the authors wrote a nonlinear equation as $$-\frac{\partial^2 \phi}{\partial t^2}+ \nabla \phi= \phi+ \sum_{k=2}^\infty g_k \phi^k$$ They have scaled the time as, $$\tau=\omega(\...
2
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0answers
565 views

Nonlinear refraction index of vacuum above Schwinger limit

This question is more about trying to feel the waters in our current abilities to compute (or roughly estimate) the refraction index of vacuum, specifically when high numbers of electromagnetic quanta ...
2
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0answers
94 views

Macroscopic chromodynamics

Lately I've been reading about gamma ray lasing phenomena, and I've been wondering about the applications of this. More concretely, the above fantastic question led me to wonder if we could somehow ...
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0answers
31 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize a dynamical system? Clearly my question looks at the same time fairly ...
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0answers
72 views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems?

In many signal processing calculations, the (prior) probability distribution of the theoretical signal (not the signal + noise) is required. In random signal theory, this distribution is typically a ...
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0answers
27 views

Good textbooks on nonlinear electrodynamics?

Looking for suggestions for a good textbook on nonlinear electrodynamics, not going into optics immediately as most textbooks tend to do but perhaps a rigorous mathematical exposition on the ...
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0answers
41 views

Non-Linear Behavior of Iterated Functional Maps

The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see http://...
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0answers
17 views

Attractors in Duffing equation

The Duffing equation in its full form is $$\ddot{x} + \delta \dot{x} -ax + \beta x^3 = \gamma \cos(\omega t)$$ Now for specific values of the parameters several attractors exist (or not). Let's ...
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0answers
102 views

Lagrangian of non-linear 3 mass, 2 spring system

Given 3 masses connected by 2 springs with the angle of intersection constant, but the springs themselves bending. Young's modulus, which is a variation of Hook's Law, applies to the flexing that ...
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0answers
17 views

How can we generate a non linear dynamical graph(equation) from nature's pattern?

Recently I was reading the scientific journal " a two dimensional network of Au nanoclusters on water surface" published in journal of nanoscience by American scientific publishers where I read how ...
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0answers
67 views

Generalized long wave and KdV equation

I have read many papers about benjamin-bona-mahony (BBM) equation or Regularized Long Wave (RLW) equation and found that BBM equation can be derived from KdV equation. from other papers i got others ...
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0answers
51 views

Origin of chaos in Chua's circuit

I am doing a project on Chua's circuit, but I can't seem to find anything that explains where the chaotic nature of the system comes from. Does anyone know of articles that explain it well on an ...
1
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0answers
25 views

Change of variables to apply Melnikov method

Supposing there is a system of non-autonomous non-linear differential equations with small damping and small forcing. The unperturbed system (zero damping and forcing) is Hamiltonian but neither has a ...
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0answers
116 views

Hysteresis in the Lorenz Equations

I was going through Strogatz's wonderful book on nonlinear dynamics and while reading through one problem he posed at the end of the chapter, I did not really understand what was going on. So I hope ...
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0answers
36 views

Data Collection of oscillatory motion

I'd like to study nonlinear oscillatory motion this semester. I plan to build several different mechanical systems (pendula, masses on springs, etc with and without driving forces, large/ small drag, ...
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0answers
95 views

Restoring force surface method for nonlinear system identification

I am working on nonlinear system parameters identification using the restoring force surface method (or the force-state mapping method). I found some references in which the method is explained but I ...
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0answers
63 views

Showing the Hamiltonian of the $\alpha$ FPU is real

I am studying the $\alpha$ FPU chain which is a model of coupled oscillators with small non-linearity. For these systems, I derived the following Hamiltonian $H$ which is given by $$ H=\sum_{j=1}^{N} \...
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0answers
153 views

Implicit Differentiation, A doubt

$v=v_c(\tau, t)$ is a smooth function and suppose we have a relation $y_c(\tau,v_c;t)=0$ when $x_c$ is written in the form $x_c=c+ty_c(\tau,v_c;t)$, $c$ is real constant, $t$ is real number denotes ...
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0answers
87 views

Toolbox for Complex Networks and Graphs

Is there a toolbox which helps in visual simulation and modeling of a network (say a mesh or ring) consisting of coupled synchronized system of nonlinear equations(ODE) which represent a system of ...
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0answers
9 views

Phase space - Diameter of attractor

For a dynamical system, with a phase space reconstructed from single scalar measurement, how do I calculate the diameter of the attractor (such as the one shown in figure below)?
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0answers
20 views

Stability of fix-point of a system of 3 non linear first order ODE, when one of the eigenvalues of Jacobian is zero

I have been working on a mean-field solution for am open quantum system model, to compare with the numerical solution of the exact solution. I have solved the system for steady state, but am now ...
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0answers
17 views

Difference between Stuart Landau equation and Ginzburg Landau equation

I have to study the Ginzburg Landau equation, but I have been told to begin by a simplier equation: the Stuart Landau one. I understand that both of these equations are used to describe nonlinear ...
0
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0answers
17 views

T-Symmetry and spatial symmetry of a multivariate conserved quantity

Definition: A reversible system is defined to be any second-order system that is invariant under the map. $t \mapsto -t$ $y \mapsto -y$ Suppose there exists a multivariate function $f(x,...
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0answers
8 views

Averaging over periodic functions in the derivation of the Kuramoto model

In the book "Chemical Oscillations, Waves, and Turbulence" Kuramoto derive his phase model. In this derivation he averaged over a fast period T (on page 66): $$ \Gamma(\psi_a - \psi_{a'}) = \frac{1}{T}...
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0answers
53 views

solutions of wave equation with cubic term

Does the following equation $$ \nabla^\mu \nabla_\mu \psi + a \psi^3 = b \psi $$ where $\psi$ is a real function, $a$ and $b$ are real constants, have other solutions that extend beyond a one ...
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0answers
10 views

What is input to state stability practically speaking

I am styuding non linear systems control and went through the input to state stability definition.Unfortunately, all I could find was some mathematical definitions. Can somebody help me to understand ...
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0answers
98 views

Second harmonic generation - how does SHG spectrum and pulse differ from the fundamental?

I'm trying to learn about second harmonic generation (SHG) in nonlinear optics but can't seem to find a conclusive answer to the following questions. 1) If generating SHG using a pulsed laser source, ...
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0answers
37 views

Nonlinear constitutive state equation

Between the Tait equation, and the B/A type of equation, which one is better suited to approximate the isentropic equation of state? Why B/A type is mostly used in nonlinear acoustics?