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14
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3answers
642 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 ...
1
vote
3answers
201 views

Solving the simplest coupled nonlinear ODES for chemical kinetics [closed]

I am just trying to get the integrated form for the kinetics of the reaction $A + B \rightarrow C + D$ characterized by: $$ -\dfrac{d[A]}{dt} = -\dfrac{d[B]}{dt} = k[A][B] \; . $$ As you note, ...
0
votes
1answer
90 views

Steady state average of physical quantities

Consider the following Hamiltonian: $$ H = \sum_n \left[\dfrac{p_n^2}{2m_n} + U(x_l) + V(x_{l+1} - x_l) \right], $$ that corresponds to a 1-D system of particles with nearest-neighbor interactions ...
1
vote
0answers
39 views

A voltage-controlled oscillator? [closed]

I already apologize for my medium english... I'm a french guy, not really gifted to recognize electronic circuits : In fact, I need to identify a circuit from is function. So, here is the block ...
3
votes
3answers
838 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 ...
2
votes
0answers
170 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 ...
0
votes
1answer
99 views

What is the difference between a linear and non-linear solution in the bending of beams?

I have been working on a simulator for bending of beams and came now to a tricky doubt: What should be the difference between a linear and non linear solution in this case (graphic at bottom)? The ...
2
votes
3answers
708 views

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 ...
1
vote
1answer
199 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 ...
1
vote
1answer
54 views

Meaning of Smooth Dynamical System?

What does smooth dynamical system mean? It is the title of a paper I am supposed to read in non linear systems.
1
vote
0answers
78 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 ...
4
votes
1answer
611 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 ...
3
votes
0answers
53 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 $X_n$ ...
3
votes
3answers
103 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 ...
6
votes
2answers
342 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 ...
4
votes
1answer
142 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 ...
1
vote
1answer
575 views

Classify equilibrium points and find bifurcation points of a non-linear dynamic system

Context: The question refers to computational physics of non linear systems with Mathematica. Excercise: Given the system $\{f_1: \dot{x} = a x + y + x^3, f_2: \dot{y} = x - y \}$: Find the ...
1
vote
0answers
22 views

Self-contained book about complex systems and nonlinear dynamics [duplicate]

I am a student at the 2nd year of a B.Sc. in Biotechnology. I started reading some papers about complex systems and nonlinear dynamics applied to economy, biology of course, climate model etc... I ...
8
votes
1answer
494 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 ...
5
votes
1answer
83 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 ...
2
votes
1answer
170 views

Bose-Einstein condensate and nonlinear waves

Can Bose-Einstein condensate be written as non-linear wave equation (in terms of mean field approximation theory)? the equation is: source: http://xxx.tau.ac.il/abs/1308.2288 What I do ...
5
votes
2answers
350 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 ...
1
vote
1answer
143 views

Why linear wave equation does not have solitonic solutions?

As many people define solitary waves they are localized pulses that propagate without changing the shape. As far as I know the same pulses exist in ordinary wave equation ! why should we look for ...
1
vote
1answer
140 views

Doubts regarding dimension of a system:Definitions and algorithms

I need to do phase reconstruction from time series data. In doing so, I encountered Takens' embedding theorem and Cao's minimum embedding dimension $d$ by nearest neighbor method. In paper "Optimal ...
2
votes
0answers
101 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, ...
3
votes
1answer
173 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?
1
vote
1answer
158 views

Dimension analysis in Derrick theorem

The following image is taken from p. 85 in the textbook Topological Solitons by N. Manton and P.M. Sutcliffe: What I don't understand from the above statement: why $e(\mu)$ has minimum ...
2
votes
1answer
47 views

Differences the nonlinerarties

I want to comparison between oscillons based on non-linearities. Can someone elaborate it with the reason behind it : When the sinusoidal vibrations are of the correct amplitude and frequency and ...
2
votes
1answer
838 views

What is the amplitude of the limit cycle of the van der Pol oscillator?

In the second edition of Classical dynamics of particles and systems by Jerry B. Marion, it is said that the van der Pol equation $$\ddot{x}-\mu\left({x_0}^2-x^2\right)\dot{x}+{\omega_0}^2x=0$$ where ...
2
votes
4answers
467 views

Nonlinear waves superposition

Non-linear waves do not superimpose to each other, but why? What characteristics give this property?
2
votes
4answers
190 views

Suggestions of a non-linear example for a small research project on numerical solution of ODEs?

I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project ...
1
vote
1answer
155 views

Finding out Energy value

A Lagrangian is given by, $$L= \left(\frac{\pi}{2}\right)^2 R^d \left[\frac{1}{2}\dot A^2 - V(A_{max})\right]$$ $$E=\left(\frac{\pi}{2}\right)^2R^d V(A_{max}) $$ where V (A) now includes nonlinear ...
0
votes
1answer
497 views

Applications to the Van der Pol equation? [closed]

What are some applications to the Van der Pol equation? Are there any physical examples?
1
vote
2answers
143 views

Linear quantization in quantum electrodynamics?

This is a continuation of this question. What would be an example of linear quantization used on quantum electrodynamics? I ask this because QED is a nonlinear theory.
1
vote
1answer
334 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? ...
6
votes
2answers
328 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 ...
3
votes
1answer
376 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 ...
1
vote
0answers
142 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 ...
5
votes
2answers
717 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 ...
4
votes
3answers
298 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? ...
7
votes
2answers
1k 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?
2
votes
2answers
305 views

Phase volume contraction in dissipative systems

I am confused about phase-volume contraction in dissipative systems. Please help me catch the flaw in my understanding. From a macroscopic point of view I understand that a dynamic system tends to go ...
4
votes
1answer
671 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 ...
2
votes
0answers
442 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 ...
0
votes
1answer
325 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 ...
-1
votes
0answers
90 views

Non-linear dynamics of classical hydrogen atom [duplicate]

Possible Duplicate: Non-linear dynamics of classical hydrogen atom I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom. ...
1
vote
2answers
270 views

Non-linear dynamics of classical hydrogen atom

I'd like to know if there have been attempts in solving the full problem of the dynamics of a classical hydrogen atom. Taking into account Newton equations for the electron and the proton and Maxwell ...
1
vote
0answers
82 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 ...
3
votes
2answers
421 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 ...
2
votes
0answers
93 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 ...