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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Supercontinoum lasers

In my work I've encountered the phrase "supercontinuoum (CW) laser" quite a bit. After reading the wikipedia page, I'm interested in a more theoretical introductory. I'm mostly interested in ...
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1answer
56 views

How to popularly describe typical features of a “non-linear system”

To quote Physics.SE tag definition: 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 ...
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No-Cloning and Uncertainty: Connections or Misconception

In chapter 9 of Scott Aaronson's book "Quantum Computing Since Democritus", he make interesting but peculiar claims relating the no-cloning theorem and the Heisenberg Uncertainty Principle (HUP). Here ...
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Why is self-organization related to scale invariance?

A lot of books mention that Scale Invariance is a property of Self-Organized critical process, but fail to mention why. Why is Scale Invariance a property of Self-Organized process.
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35 views

Nonlinear Saturated Schrodinger Equation in 1D- Physical Models

I'm studying the Nonlinear 1d Schrodinger equation $$i\psi _t + \psi '' + |\psi |^{2p} \psi - \epsilon |\psi | ^{2q} \psi = 0\, , \quad t>0, x\in \mathbb{R}\, ,$$ and specifically, its solitary ...
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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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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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1answer
91 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 ...
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2answers
109 views

Validity of the Lyapunov exponent approximation

I was trying to get the Lyapunov exponent for some dynamical nonlinear systems and found that it is not true (as I had expected) that the distance between two trajectories with slightly different ...
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1answer
39 views

Solution of Burgers' equation in preshock region

According to Hamilton's & Blackstock's Nonlinear acoustics (Section 4.5.4) the solution of Burgers' equation of the form: $$ \frac{\partial P}{\partial \sigma} - \frac{1}{\Gamma}\frac{\partial^2 ...
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1answer
40 views

What kind of damping is this $F = -ax|x'|$?

From Applied Mathematics by Logan: A mass hanging on a spring is <...> governed by $$mx'' = -ax|x'| - kx$$ where $-ax|x'|$ is a nonlinear damping force. I looked up "nonlinear damping" ...
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21 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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Is it possible to calculate the Hamiltonian for the Kuramoto model ? If yes, then how? [duplicate]

I just want to know whether the Hamiltonian for the Kuramoto model is possible or not.Also does the temperature have any affect on its synchronization or not? Thanks in advance.
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139 views

How to properly use Perturbation Theory in classical systems?

Context: If we consider a particle in upwards motion near the Earth's surface and acted by a quadratic drag we get the non-linear eom: $$\frac{dv}{dt}=-g-\frac{b}{m}v^2.$$ We can solve it ...
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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 ...
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18 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 ...
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2answers
65 views

Question about a Attractors in Non-linear Systems

I've recently been reading up on non-linear dynamics and came across the concept of attractors. I'd like to ask if the concept of attractors can be used for pedestrian egress from a room? Since ...
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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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1answer
75 views

Homoclinic orbit and a particle in a double well

The physical set-up is a classical particle in a parabolic double well: Physically, a particle with reasonable amount of potential energy would be able to roll down the slope of the well, roll past ...
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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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74 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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1answer
49 views

Failure of Superposition principle at high amplitudes

Why does superposition principle fail at high amplitudes. Please answer with respect to transverse waves. If possible, plane progressive transverse waves at best.
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1answer
21 views

Is there nonlinear system have both stable and asymptotically stable equilibrium points?

A nonlinear dynamical system can have multiple equilibrium points with different characteristics. I know that a pendulum with friction model $$\dot x_1 = x_2$$ $$\dot x_2 = -\dfrac{Mgl}{I} \sin(...
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1answer
105 views

Symbolic dynamics of a multidimensional system

Let $x_t = F(x_{t-1})$ be a discrete-time dynamical system in the chaotic regime. Starting from an initial condition $x_0$, we can generate a time series $(x_t)$ where $t =1,2,...,T$ indicates the ...
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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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Why can some oscillations be modeled by Simple Harmonic Motion, while others cannot?

For some oscillators an increase in the driving amplitude changes the period (frequency) of the oscillation, but the simple harmonic oscillator does not predict this type of behavior. Why?
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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 ...
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57 views

Poincare-Bendixson Theorem Under Time Reversal

Strogatz's textbook "Nonlinear Dynamics and Chaos", Chapter 7 presents the Poincare-Bendixson theorem, which gives conditions under which one can conclude the existence of a closed orbit within some ...
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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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2answers
48 views

Question about limit cycles and linear systems

In here http://users.isy.liu.se/en/rt/claal20/SysBio2015/Notes_SysBio_2015_partC.pdf it says: A limit cycle is however an intrinsically nonlinear concept: a linear system cannot have a limit ...
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62 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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2answers
46 views

Why do materials show plastic behaviour for large stress?

As the stress is increased, the strain increases proportionally up to elastic limit and the material regains its original dimension within elastic limit. When the stress is increased further the ...
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1answer
62 views

Do all equillibrium points of a discrete mapping show up on the bifurcation diagram?

The question in the title is perhaps vaguely posed, so I'll include the concrete example which is bugging me. Suppose we have a mapping given by $$N_{t+1}=N_t\cdot \exp(r(1-N_t-PN_t/(\alpha^2+N_t^2)))...
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What explanation can we give for the generation of spiral waves in a excitable medium?

I was thinking about the reason for the generation of spiral waves (a.k.a scroll waves) like in BZ reactions and Fitzhugh-Nagumo systems. Can someone give me some explanation or references ?
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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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1answer
59 views

Can a dynamical system have an infinite critical points? [closed]

I have studied the cosmological evolution of dark energy modeled as a scalar field. I want to make an extension to link and I have arrived at a system of differential equations on the following form $...
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8answers
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Turbulent spacetime from Einstein equation?

It is well known that the fluid equations (Euler equation, Navier-Stokes, ...), being non-linear, may have highly turbulent solutions. Of course, these solutions are non-analytical. The laminar flow ...
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1answer
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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 ...
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1answer
45 views

Why trajectories approach to origin tangent to the slower direction?

I am reading non-linear dynamics from Strogartz. Suppose, I have two solutions of a non linear system: $x(t) = x_0e^{at}$ and $y(t) = y_0e^{-t}$, where $a\in \mathbb{R}$. Now it is clear that,for $a&...
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Given force as function of position, find the total energy as function of time [closed]

Given that the force for a non-linear spring connected to a mass $m$ sitting on a table is $$f(x) = -kx -ax^3,$$ Find the total energy as a function of time $E(t)$. I have no clue where to begin.
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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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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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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 $I(x)=I(f(x))$. ...
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1answer
145 views

Analytic proof that Lyapunov exponents in Hamiltonian systems pairwise sum to zero

I have read that in Hamiltonian systems, Lyapunov exponents come in pairs $(\lambda_i, \lambda_{2N-i+1})$ such that their sum is equal to zero. Is there a way of proving this analytically? EDIT: ...
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12 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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101 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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70 views

Why do spiral waves annihilate each other when 2 wavefronts collide?

I was reading about Fitzhugh-Nagumo model. And in a 2D space the simulations a Reaction-Diffusion process associated with FitzHugh system look like this. But intuitively I could not satisfy myself ...
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1answer
51 views

In reaction-diffusion processes what is the difference between oscillatory media and excitable media?

What is the basic differences between oscillatory media and excitable media? I know that both comes under reaction-diffusion processes. Where do Turing patterns come in the picture? Can some one give ...
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94 views

When is the phase space diagram an ellipse?

For a two dimensional dynamical system, when does the phase space diagram give an ellipse? I know about the examples for damped and undamped harmonic oscillators, but our instructor said that the ...
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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{...