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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Explanation of the waves on the water planet in the movie Interstellar?

We will ignore some of the more obvious issues with the movie and assume all other things are consistent to have fun with some of these questions. Simple [hopefully] Pre-questions: 1) If the water ...
3
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36 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
24 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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A good, concrete example of using “chaos theory” to solve an easily understood engineering problem?

Can anyone suggest a good, concrete example of using "chaos theory" to solve an easily understood engineering problem? I'm wondering if there is a an answer of the following sort: "We have a high ...
3
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1answer
29 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 ...
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2answers
136 views

Why is my Lyapunov exponent similar for single and double pendulum?

This is my first question here on stackexchange. I hope that I can be understood. If not, tell me and I will reformulate and fill in with details. I have simulated a single pendulum and a double ...
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1answer
225 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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41 views

How to do linear stability analysis on this system of PDEs?

I was reading this paper. The model as in the paper is given below. Is it possible to do a linear stability analysis on this system? If so can someone help me?
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2answers
281 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 ...
0
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2answers
45 views

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 ?
8
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2answers
281 views

Chaos is predictable?

I'm reading a book of computational physics [1] where the driven nonlinear pendulum is studied in depth. This is the equation derived in the book: $$ \frac{d^2\theta}{dt^2} = -\frac{g}{l}\sin\theta - ...
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32 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 ...
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1answer
41 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 ...
55
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8answers
5k views

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 ...
5
votes
1answer
69 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 ...
0
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1answer
43 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 ...
0
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2answers
60 views

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 ...
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14 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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48 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 ...
4
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1answer
92 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: ...
3
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2answers
178 views

Infinitesimal input, macroscopic output

I must admit that I never got well how physicists handle infinitesimal quantities, mainly because of my education as a mathematician. So the following lines (taken from the preface of Berezin and ...
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38 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 $I(x)=I(f(x))$. ...
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53 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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9 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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1answer
55 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 ...
0
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1answer
41 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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1answer
82 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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121 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) + ...
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108 views

What are the differences between logistic map, poincaré map, attractor, phase portrait, bifurcation diagram? [closed]

What are the differences between Logistic map, Poincaré map, Attractor, Phase portrait, Bifurcation diagram Currently I became interested in chaos theory and non-linear dynamics. While ...
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0answers
81 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
22 views

Is there any book about chaos theory and nonlinear dynamics? [duplicate]

I'm interested in chaos theory and nonlinear dynamics. I learned some knowledge about phase space, attractors, bifurcation diagram, etc from Wikipedia. But I want to study more comprehensive about ...
0
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1answer
59 views

what is the source of nonlinear behaviour of semiconductor transistor?

It is known that semiconductor transistor can be used as a switch as well as amplifier. How does the linear amplifying characteristic and non-linear switching characteristic come in a single system?
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31 views

What determines the point of energy spillover to higher modes of a standing wave resonator?

One of the better known physics demonstrations for standing wave resonance is the singing rod . By holding the rod exactly in the middle the demonstrator constrains the first mode of excitation - the ...
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1answer
487 views

Explain the Föppl–von Kármán equations

I am a newbe to elasticity. Could someone please explain to me briefly how the Föppl–von Kármán equations work? What are we trying to solve for? Is there some kind of intuition to the way they look? ...
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3answers
3k views

Why can't the Navier Stokes equations be derived from first principle physics?

At the 109th UCLA Faculty Research lecture, Seth Putterman gave a talk on Sonoluminescence. During the lecture he emphasized that "The Navier Stokes equations cannot be derived from first principles ...
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388 views

Why are non-linear optics called non-linear?

Looking at the wikipedia article on nonlinear optics you can see a huge list of frequency mixing (or multi-photon) processes. What makes these different from single-photon interactions? More ...
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343 views

Modeling stochastic process with frequency-dependent power spectrum

I'm trying to model of Johnson-Nyquist noise propagation in a nonlinear circuit. An ideal (linear) resistor can be modeled very nicely by the Fokker-Planck equation (equivalently, the drift-diffusion ...
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4answers
114 views

Energy conservation without action principle?

The normal tagline for energy conservation is that it's a conserved quantity associated to time-translation invariance. I understand how this works for theories coming from a Lagrangian, and that this ...
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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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59 views

What does the term 'hyperbolic model' mean?

I am reading this non-linear discrete dynamical system paper. The authors mention the term hyperbolic model. What does that mean?
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107 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 ...
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1answer
30 views

To describe crystallity structures of this ferromagnetic material

MOKE microscope picture of the ferromagnetic Material $Co_{40} Fe_{40} B_{20}$ of 20 nm thin film All other pictures look the same, also from different angles: [0,360] by 15 degree separation. I ...
3
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1answer
339 views

Nonlinear waves and shock formation

In the cases of nonlinear acoustics, why is shock formation unlikely when the dispersion is strong when compared to the nonlinearity of the wave?
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695 views

Why is the computer useful if a chaotic system is sensitive to numeric error?

In every textbook on chaos, there are a lot of numerical simulations. A typical example is the Poincare section. But why is numerical simulation still meaningful if the system is very sensitive to ...
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56 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 ...
0
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1answer
93 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 ...
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47 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 ...
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Current scope of Chaos theory and non-linear dynamics? [closed]

I am a physics undergrad interested in stuff like dynamical systems, chaos theory etc. Is there ongoing research in these fields? I am talking about pure research and not applications to things like ...
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22 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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376 views

Inhomogenous Schrödinger equation

Please help me out in solving this inhomogeneous Schrödinger equation in cylindrical co-ordinates [You may suggest if I have to go for mathematics]: $$ \ddot R + \frac1r\dot ...