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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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 ...
26
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3answers
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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 ...
14
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3answers
712 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 ...
14
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3answers
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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 ...
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4answers
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Non-linear systems in classical mechanics

In general, what is meant by non-linear system in classical mechanics? Does it always concern the differential equations one ends up with (any examples would be greatly appreciated)? If so, is it ...
9
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994 views

Could the Schrödinger equation be nonlinear?

Is there any specific reasons why so few consider the possibility that there might be something underlying the Schrödinger equation which is nonlinear? For instance, can't quantum gravity (QG) be ...
9
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2answers
282 views

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 ...
9
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2answers
1k views

warp drive with gravitational waves in the nonlinear regime

gravitational waves are strictly transversal (in the linear regime at least), also their amplitudes are tiny even for cosmic scale events like supernovas or binary black holes (at least far away, ...
8
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2answers
293 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 - ...
8
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1answer
931 views

Anharmonic oscillators: why is $F=-k x-k' x^3$, with no quadratic terms?

The equation of motion of a general anharmonic oscillator includes a position-dependent force that can be expanded in a Taylor series as $$m\ddot{x}+2\mu\dot{x}+k_0+k_1x+k_2x^2+k_3x^3\ldots=F.$$ I ...
8
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3answers
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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 ...
8
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1answer
165 views

Deviation from power law distribution of earthquakes

One of the most accepted frameworks for the relationship between the magnitude and frequency of an earthquake is that of the critical phenomena. In this framework, the magnitude of events must be ...
7
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2answers
2k views

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics?

Does chaos theory occur in quantum mechanics? Or in any non-newtonian physics? Apart from perhaps thermodynamics?
7
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2answers
1k views

Nonlinear dynamics beneath quantum mechanics?

Yesterday I asked whether the Schroedinger Equation could possibly be nonlinear, after reviewing the answers and material given to me in that thread I feel like my question were adequately answered. ...
6
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1answer
3k views

Poincaré maps and interpretation

What are Poincaré maps and how to understand them? Wikipedia says: In mathematics, particularly in dynamical systems, a first recurrence map or Poincaré map, named after Henri Poincaré, is ...
6
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4answers
115 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 ...
6
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2answers
350 views

How to find zero-point oscillations for this system?

Consider the following Hamiltonian which is absolutely relativistic literally: only sensitive to absolute pairwise relative phase space variables of objects for a system of $N$ objects moving in one ...
6
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2answers
342 views

Change of basis in non-linear Schrodinger equation

At the mean-field level, the dynamics of a polariton condensate can be described by a type of nonlinear Schrodinger equation (Gross-Pitaevskii-type), for a classical (complex-number) wavefunction ...
6
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2answers
297 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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2answers
557 views

How and why can random matrices answer physical problems?

Random matrix theory pops up regularly in the context of dynamical systems. I was, however, so far not able to grasp the basic idea of this formalism. Could someone please provide an instructive ...
5
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2answers
1k views

What is a physical example of a Saddle-Node Bifurcation?

I am doing a presentation on bifurcations and would like physical examples to go along with each type of bifurcation but I am unable to find or think of any good example of a simple Saddle Node ...
5
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2answers
347 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 ...
5
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2answers
529 views

Does the logistic map have an attractor for a particular value of the parameter?

Background: Currently I am studying a course on non-linear dynamics. We have been studying about attractors only intuitively, so I do not have a definition for an attractor. Let me give you a couple ...
5
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1answer
91 views

Stability theory [closed]

I'm studying stability theory recently and met a lot of phrases like linear stability and nonlinear instability. After searching on Google, I became more confused. Thus I wonder if there is any ...
5
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1answer
71 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 ...
4
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3answers
351 views

Non-SHM oscillatory motion

How to solve these kind of questions , where $|F| \propto x^2$? How to find time period and velocity type related things to the oscillatory motion? ...
4
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3answers
2k views

Liouville's theorem and conservation of phase space volume

It can be proved that the size of an initial volume element in phase space remain constant in time even for time-dependent Hamiltonians. So I was wondering whether it is still true even when the ...
4
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2answers
403 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 ...
4
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2answers
471 views

What are the solution spaces of Nonlinear Schrödinger equations?

As we know, the solution space of Schrödinger equation is a Hilbert space, however, what about it of Nonlinear Schrödinger equations such as $$i\partial_t\psi=-{1\over ...
4
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1answer
122 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: ...
4
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1answer
341 views

Viscous Burgers equation physical meaning

The viscous Burgers' equation: $$ q_{t}+q\:q_{x}~=~\nu\:q_{xx}, \mbox{ where } \:\:\nu >0, $$ combines the nonlinear propagation of $q(x,t)$ and the diffusion. What is this equation for? (in ...
4
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1answer
849 views

Gross-Pitaevskii equation in Bose-Einstein condensates

I was hoping someone might be able to give a approachable explanation of the Gross-Pitaevskii equation. All the sources I've been able to find seem to concentrate on the derivation, and I don't have ...
4
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1answer
827 views

Calculating Lyapunov exponents from a multi-dimensional experimental time series

Wolf's paper Determining Lyapunov Exponents from a Time Series states that: Experimental data typically consist of discrete measurements of a single observable. The well-known technique of phase ...
4
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1answer
156 views

How to find positions of $n$ masses in Newton mechanics?

I ran into a problem while doing research. The problem can be described as: consider the original $n$-body problem, and if we fix the position of them(unknowns), no interaction among them, they don't ...
4
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1answer
52 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 ...
4
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1answer
156 views

Why is the critical state in the Mean Field Bak-Sneppen Model a global attractor?

Simulations of the Bak-Sneppen Model of Species Evolution (introduced in http://link.aps.org/doi/10.1103/PhysRevLett.71.4083) show that it exhibits Self-Organized Criticality where after a transient ...
4
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0answers
143 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) + ...
4
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0answers
49 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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1answer
428 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?
3
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2answers
180 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 ...
3
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3answers
115 views

Simple Experimental Laser Characterization Parameters Ideas?

I have 2 months to do a project on characterization of a laser parameter. We have typical optics lab equipment (DFB Lasers, oscilloscopes, random number generator, etc.). I was told to choose a laser ...
3
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1answer
48 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 ...
3
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1answer
471 views

The universality of the Stuart-Landau equation to describe nonlinear oscillators

I have read numerous papers which boldly suggest that the Stuart-Landau equation can be successfully used to model any weakly nonlinear oscillating system near a Hopf bifurcation. Even thought it has ...
3
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3answers
692 views

What is a nonlinear field?

I have read two possible definitions. A nonlinear field is A field taking values on a manifold. A field whose equation is nonlinear. What do you understand by a nonlinear field or a nonlinear ...
3
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2answers
189 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 ...
3
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1answer
560 views

Deflection of a membrane

I am currently working on a project which is described as the deflection of a circular membrane. What I am trying to model is the deflection of a piece of plastic film (E=200MPa,v=0.5) when placed ...
3
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1answer
57 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 ...
3
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0answers
90 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 ...
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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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3answers
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Non-linear dynamics vs Chaos

I am confusing between non linear dynamics and chaos. Chaos is also a non-linear dynamics right? then what is the difference between chaos and non-linear dynamics? What I understood about chaos is ...