All Questions
Tagged with non-linear-dynamics or non-linear-systems
473 questions
4
votes
2
answers
213
views
Why would we want to calculate the Lyapunov exponent for experimental data?
Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even ...
- 41
4
votes
2
answers
1k
views
How are jerk equations connected to chaos theory?
I read in this Wikipedia article:
It has been shown that a jerk equation, which is equivalent to a system of three first-order, ordinary non-linear differential equations, is the minimal setting ...
0
votes
3
answers
495
views
What are the necessary and sufficient conditions for a motion to be periodic?
Consider the following idealized motions
(i) The motion of a bob attached to a spring on a horizontal frictionless table,
(ii) the motion of a pendulum with an arbitrary amplitude without air ...
- 12.3k
6
votes
1
answer
669
views
Physical meaning of third derivative with respect to position
I currently on a numerical solver for the KdV equation which reads
$$ u_t + uu_x = u_{xxx} $$
I was wondering the physical sense of this third derivative with respect to $x$. I know that the $uu_x$ ...
user252548
1
vote
0
answers
64
views
Instability of coupled non-linear oscillators
Consider a bunch of interacting oscillators (e.g., a chain of atoms), interacting due to anharmonicity in the potential energy. You can Taylor expand the force on each oscillator about equilibrium ...
- 1,772
1
vote
1
answer
142
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Is the quantum dynamics of a system of interacting particles linear or non-linear?
As I understand it, the linearity of quantum mechanics is considered to be an inviolable principle - e.g., this paper - because (among other things) causality would be violated or and/or superluminal ...
- 25k
1
vote
0
answers
74
views
What can one conclude about the stability of limit cycles without the use of numerical methods?
Let's assume one asserts the existence of a closed orbit by applyling the Poincaré-Bendixson theorem to a trapping region $R$ that is constructed such that all phase vectors on its boundary point ...
- 61
-2
votes
1
answer
132
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Lorentz Equation Symmetry
I was going via Lorentz equation & learning the topic on Symmetry, what I couldn't understand is how did they performed this type of substitution & what is the philosophy behind this way of ...
- 107
2
votes
0
answers
88
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Calculating the absorption cross-section for Helium in a strong oscillating external IR field -> AC Stark Shift (Autler-Townes Splitting)
I try to obtain the absorption cross-section for atomic helium, in a strong and oscillating IR field, for when a second - XUV - pulse is probing the system. I think the only way of doing this is to ...
2
votes
0
answers
172
views
Gravitational non-linearities and Dark Matter/Energy
I had read Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies (https://arxiv.org/abs/1909.00095), was wondering
Is there good reason to think that gravitational non-...
- 112
4
votes
2
answers
366
views
Is there anything more chaotic than fluid turbulence?
Fluid turbulence is a highly complex and non-linear chaotic phenomenon. Great difficulties and complications are encountered when trying to accurately and robustly calculate or simulate fluid flows, ...
- 4,064
0
votes
1
answer
68
views
Could different outcomes have different physics in Wigner's friend?
Could different outcomes have different physics in Wigner's friend?
Physicist Eugene Wigner said that consciousness was fundamental for physics and that laws of physics existed because of it. He said ...
- 2,868
1
vote
1
answer
150
views
Intuition behind focusing vs defocusing in integrable systems like NLS, KdV, mKdV
The following are examples of integrable systems arising from the AKNS system (check out AKNS paper here and a short Wikipedia description)
Non-Linear Schrodinger equation
Korteweg-de Vries equation
...
- 11
0
votes
1
answer
134
views
Natural phenomena with cubic behaviour [closed]
I'd like to know which natural phenomena (in planet earth) may be described with a cubic function/polynomial? or is there not any.
Accelerated movement is quadratic. Work, is also quadratic. the ...
- 101
0
votes
0
answers
52
views
Dimension of the constant in Born-Infeld nonlinear electrodynamics
As I know, based on the Lagrangian of Born-Infeld electrodynamics, its constant which shows the strength of electromagnetic field should have the dimension of inverse of length, but in some papers I ...
- 35
0
votes
0
answers
63
views
How to get the principle stretches of compressible Neo-Hookean material under uniaxial extension?
As the title described, how the principal stretches of a compressible Neo-Hookean material undergoing uniaxial extension are derived from the constitutive model as below?
$$
\lambda_1 = \lambda; \...
- 1
1
vote
1
answer
164
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Minimal dynamical system with quasiperiodic oscillations
What is a minimal, explicit dynamical system (as in, a series of coupled ordinary differential equations) that exhibits quasiperiodic oscillations for some region of parameter space? Two coupled Van ...
- 235
2
votes
1
answer
216
views
The solution to the non-linear convection equation
The non-linear convection equation
$$u_{t} +uu_{x}=0$$ admits implicit solutions of the form $$u=f(x-ut).$$
How does one interpret this solution intuitively? Is there an example of a solution of this ...
- 171
2
votes
1
answer
2k
views
How is a quartic oscillator solved in classical mechanics?
Quantum mechanically, a quartic anharmonic oscillator with potential $$V(x)=\frac{1}{2}m\omega^2x^2+\lambda x^4$$ is dealt with perturbation theory- the approximate energies $E_n$ and energy ...
- 12.3k
-1
votes
1
answer
271
views
How do we find the equation for the gyrating motion of a particle in a uniform magnetic field and a non-uniform Electric field? [closed]
Considering the gyrating motion is not negligible and also retaining the guiding center drift, how do we get the trajectories x(t),y(t),z(t) of the particle?
In this case is the variation in the ...
- 63
0
votes
1
answer
49
views
In what sense do bifurcations concern change in quality?
I've heard such vague statements several times and also read:
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family.
(From ...
- 21
2
votes
1
answer
433
views
Why does a non-linear system lead to interaction and frequency mixing between inputs?
When we have a system that is nonlinear and we apply a sum of two different frequency sine waves as an input, we see the output of this system has components that are at the sum frequency of the two ...
- 89
1
vote
1
answer
69
views
What is the general definition of thickness of a strange attractor?
Disclaimer: This question is cross posted on Math.SE because I don't know which site is more appropriate for this question.
In Chaosbook, at page 56, it is asked to find the thickness of Rössler ...
- 2,313
12
votes
1
answer
750
views
What quantum phenomena violate the superposition principle in electromagnetism?
On page 11 of the 3rd edition of Electricity and Magnetism by Edward M. Purcell and David J. Morin it says:
"we know of quantum phenomena in the electromagnetic field that represents a failure of ...
- 289
1
vote
2
answers
558
views
How to calculate the parameter values for which the Lorenz system is chaotic?
I was recently going via a book (Strogatz), that mentions Lorenz's attractor, and that it was found out that for values such as $a=10$, $b=\tfrac{8}{3}$, $c=21$, the system behavior is chaotic.
How ...
- 107
1
vote
1
answer
166
views
Poincaré Map (Quasi-periodicity; Stability)
In a Poincaré map, when quasi-periodicity is exhibited by the dynamical system, what does it mean in terms of stability for the dynamical system?. Why is it so that as Maximum Lyapunov exponent (MLE) ...
- 107
1
vote
0
answers
299
views
Non-quadratic kinetic energy [closed]
Do you have examples of Lagrangians/Hamiltonians used in physics with non-quadratic kinetic terms? e.g. $\dot{x}^4$
What is the origin and the interpretation of such terms?
- 1,231
1
vote
0
answers
49
views
Is there any nonlinear equations depending on Fourier coefficients?
A nonlinear partial differential equation is an expression depending on derivatives of $u$
$$f(x,t,u,u_x,u_t,\cdots)=0,$$
where the derivatives of $u$ can be obtained from the Taylor series of $u$.
...
- 111
2
votes
1
answer
454
views
Vacuum birefringence
Many of the papers (e.g., this) dealing with nonlinear electrodynamics treat a theory's prediction of vacuum birefringence as undesirable, but don't explain why it would be undesirable. For example:
...
- 25k
4
votes
1
answer
466
views
Intuition behind the meaning of Lyapunov exponents
Can anyone help me in understanding the contraction and the expansion of the phase space? what are Lyapunov exponents? and how come one understand this concept intuitively?
- 107
0
votes
0
answers
35
views
Distinguishing a LTI from not with unknown inputs
Linear time invariant (LTI) systems are a staple of physics. They appear in many situations. But how do you know a system is a LTI?
In particular, if you are provided with a black box which ...
- 51.7k
0
votes
0
answers
64
views
Infinite series vs compact representation
I understand the attractiveness and usefulness of infinite-series expansions such as Taylor expansions, but I wonder if they sometimes hide important aspects of the described system.
For example, ...
- 25k
0
votes
1
answer
233
views
What does the phase discriminator portion of the Costas Receiver do mathematically?
What does the phase discriminator portion of the Costas Receiver do mathematically?
The output of the $I$-channel is $ \frac{1}{2}A_C \cos \phi \, m(t) $. Which means for small deviation of phase $ \...
- 33
0
votes
1
answer
391
views
Solution of the coupled non-linear oscillators by using perturbation theory [closed]
The integration shown here, $$∫_{-\infty}^{+∞}x^r\mathrm{Exp}[−x^2]\mathrm{H_n}^2[x]\mathrm{d}x,$$ appears when we try to calculate the spectrum of the perturbed non-linear oscillators by using ...
- 61
-1
votes
3
answers
99
views
Solving ODE equation for classical field [closed]
I would like to solve the following homogeneous, ODE:
$$\left[\frac{d^2}{dt^2} + m^2\right]\phi(t) + \frac{1}{6}\lambda \phi^3(t)=0.$$
I know the solution is
$$\phi(t) = \frac{z(t)}{1-\frac{\lambda}...
- 357
0
votes
0
answers
72
views
Temperature distribution and evolution in a greenhouse
I am trying to mathematically model the temperature distribution and evolution in a greenhouse, which conserves heat due to the greenhouse effect. Here is a transverse schematic ($H$ is the height of ...
- 11
0
votes
1
answer
98
views
Characteristics of acoustic resonator with a constant gain frequency response
What would be the theoretical characteristics of an acoustic resonator cavity which has a completely flat gain frequency response over 200Hz-3000Hz (Roughly the range of a violin)
In other words, ...
- 103
2
votes
0
answers
335
views
Exact solution for non-linear Fokker-Planck equation
I'm searching for exact (analytical) results for FP equation in 2 variables (such as $x$ and $p$ in 1D) with a steady state. Kramer's like (with force due to confining potential, such as harmonic ...
- 2,342
2
votes
1
answer
666
views
Poincaré plane and Logistic Map
How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
user212422
12
votes
5
answers
4k
views
Is non-linear quantum mechanics possible?
Say we have a state vector $|A\rangle$. Is it possible to have a theory where the evolution of $|A\rangle$ depends on the vector $|A\rangle$ itself? e.g.
$$ i\frac{\partial}{\partial t} \psi(t) = \...
user84158
3
votes
2
answers
663
views
Poincare return map as area-preserving map
I'm trying to get some intuition into how the Poincare return map is area-preserving (when there are two momenta and two positions).
Suppose $H=H(q_1,q_2,p_1,p_2)$, and let's suppose the system is ...
- 47.8k
1
vote
1
answer
175
views
Unpredictability, per definitions of chaotic behavior
Apparently I've been confused about the meaning(s) of "chaotic behavior". I always thought it meant that infinitesimal perturbations of a system parameter would lead to large changes in the system's ...
- 25k
6
votes
1
answer
1k
views
Is *every* autonomous first-order planar/2D system integrable?
Consider a general autonomous first-order planar/2D system:
$$\begin{cases}
\frac{dx}{dt} = A(x,y)\\
\\
\frac{dy}{dt} = B(x,y),
\end{cases}$$
where $A,B$ are two functions. Reading Classical Mechanics ...
- 647
0
votes
1
answer
90
views
quantized energies for a particle in a non-linear potential
Okay, so the question i'm trying to solve is to find the quantized energies for a particle in the potential:
$$V(x)=V_0 \left ( \frac{b}{x}-\frac{x}{b} \right )^2$$
for some constant b. I used the ...
- 29
1
vote
2
answers
223
views
Sum harmonic and sum frequency generation
Can two collinear beams of two different wavelengths generate the sum frequency or they need to pass each other at a certain angle. For a monochromatic light how does the sum harmonic gets generated ...
- 43
0
votes
0
answers
125
views
Two E-fields and two energy levels create infinite frequencies?
In this paper it says that for a two-level system excited by two fields:
$$ V_{ab} = -\mu_{ba} E_1 e^{i \omega_1 t}+ E_3 e^{-i \omega_3 t}$$
"In steady state the off-diagonal density-matrix ...
- 2,281
3
votes
2
answers
153
views
Do meaningful bifurcation diagrams exist for systems described by vector fields on circles?
I've been reading about the vector field on a circle, and how it's used to describe stable points for periodic motion. I have also read about how bifurcation diagrams describe changes in positions of ...
user191954
1
vote
1
answer
434
views
What does that means? "QCD is a non-linear and non-trivial field theory?"
I know QCD is represented by the $SU(3)$ group and is non-abelian. Then, as a consequence QCD is a non-linear and non-trivial field theory.
I would like to know why? and what does that means?
- 11
2
votes
1
answer
266
views
Finding dispersion relations
I was wondering if there is a general (theoretical, not experimental) method for finding the dispersion relation for waves in a medium, say given the equation governing purturbations in the medium? ...
- 4,067
2
votes
1
answer
95
views
Entropy of natural networks [duplicate]
How does one define the entropy of a natural network (say for example, a river network, or a morphological skeletal network of a lake in the figure below) ? For example, the following report suggests ...
- 425