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4
votes
1answer
151 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 ...
11
votes
4answers
390 views

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 ...
1
vote
0answers
19 views

Fixed points of: $\dot{x}=\sin(y)\\ \dot{y}=\cos(x)$ [migrated]

How can you find the fixed points of this system: $\dot{x}=\sin(y)\\ \dot{y}=\cos(x)$ Normally I would suggest that you find the points when both functions are equal to 0.
0
votes
0answers
24 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?
0
votes
0answers
71 views

Pdf of particles in phase space

I am from a non-physics background and hence have difficulty in grasping the concepts some of which I need to apply in signal processing. Link1 and Link2 discuss about Liouville theorum and phase ...
1
vote
0answers
55 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} ...
2
votes
0answers
22 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, ...
1
vote
0answers
62 views

Phase space volume and correlation dimension

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
2
votes
3answers
68 views

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

I was trying to do linear stability analysis of spring pendulum. I arrived at the differential equations which describe the system. But I am unable to proceed to linear stability analysis. Is it ...
1
vote
0answers
56 views

Autocorrelation function for deterministic nonlinear dynamical systems

I am quite puzzled with the problem that spectral analysis has been either applied to noisy dynamical systems or to chaotic ones. I was wondering why nobody makes analysis of non-linear dynamical ...
3
votes
0answers
45 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 ...
0
votes
0answers
15 views

How far is the range of an ultrasonic sensor as far as it can transmit? [duplicate]

How far can ultrasonic waves propagate in the air? (if the energy is enough), how far can ultrasonic waves propagate in the ground (concrete floor)? Are there any articles or experiments about ...
0
votes
0answers
14 views

When longitudinal waves and transversal waves enter air from a soild, what will happen? How do waves transform?

When ultrasonic waves meet the interface (surface) between two media they will reflect and refract. What is the effect of the acoustic impedance on reflection? Is there any list about acoustical ...
2
votes
0answers
48 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. ...
4
votes
0answers
37 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 ...
1
vote
1answer
61 views

Is there a normalized form of the Euler equation discretized with finite volumes?

I want to calculate a flux on my fpga using the Euler equations with the finite volume method. Unfortunately the values of the state variables differ a lot. For example the pressure has a value of ...
3
votes
1answer
61 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 ...
0
votes
1answer
35 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?
14
votes
2answers
553 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
124 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
71 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
28 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 ...
2
votes
2answers
328 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 ...
1
vote
0answers
128 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
53 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
435 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
156 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
51 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.
0
votes
0answers
47 views

What is a non-linear capacitor?

I'm new here. I have a (maybe dumb) question: What are non-linear capacitors? I'm given a circuit including a capacitor and the question says The given capacitor is non-linear with the ...
1
vote
0answers
55 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 ...
0
votes
0answers
33 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
4
votes
1answer
320 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
49 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 $X_n$ ...
0
votes
0answers
38 views

Experimentally determining photon lifetime of a laser from transient response

At our lab, we have a DBR laser set up to some measuring equipment (oscilloscope, spectrum analyzer and random number generator). I have been tasked to experimentally obtain the photon lifetime of the ...
3
votes
3answers
81 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
322 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
135 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 ...
0
votes
0answers
27 views

Pick marginal circles in phase plot of a non-linear dynamic system

Context: I'm studying non-linear dynamics with Mathematica. Part of the problem: Given the following system: $\ddot{x} = x - x^3 - 0.2 \dot{x} + g(\sin(t) + \cos(2t))$, find two values of $g$ that ...
1
vote
1answer
332 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
13 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
268 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
75 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
138 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
228 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
112 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
111 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
96 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
89 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
106 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 ...
1
vote
1answer
36 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 ...