All Questions
Tagged with non-linear-dynamics or non-linear-systems
473 questions
2
votes
1
answer
418
views
Equivalent system in Centre manifold theory
I was studying the centre manifold theory. It says (see Kuznetsov (pdf) page 155, theorem 5.2) that the system on the left side of the picture is topologically equivalent to the one on the right.
$
\...
- 1,424
2
votes
1
answer
277
views
Phase space portrait for dynamical system with Bifurcations
I have this dynamical system
$$x'=y, y'=-x^3-y+mx$$
and I want to draw the phace space diagram for $m=-1/8, m=1/4,$ the bifurcation points. 1st of all I cant find what kind of bifruction I have( I go ...
- 2,566
4
votes
2
answers
2k
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 ...
- 11
0
votes
0
answers
80
views
Non-linear Diffusion Equation
I'm currently trying to solve the equation $$
\frac{\partial C}{\partial t}= \frac{\partial}{\partial x}\left(\frac{D}{C}\frac{\partial C}{\partial x}\right),
$$ where D is a constant and $C \equiv C(...
- 35.5k
1
vote
2
answers
105
views
How to draw the phase plane of this equation?
Using various computational tools, it's possible to draw a phase plane from two first-order ODEs or a single second-order ODE. However, when there is a parameter in the equation and we don't know the ...
- 66.3k
1
vote
1
answer
68
views
Mixing for Burgers equation in 2+1D
Let us consider the following (2+1)-dimensional Burgers-like equation:
$$
u_t + (u^2)_x + (u^3)_y=0.
$$
Here the unknown is a function $u= u(t,x,y):(0,\infty) \times \mathbb R^2 \to \mathbb R$.
Is ...
5
votes
1
answer
602
views
Adjoint of a non-linear operator
I am a retired aerospace engineer, embarking on a self-study of QM. In reading S. Weinberg's book Lectures on QM (second ed.) I found the following definition (pag.65):
"The adjoint $A^\dagger$ ...
- 213k
1
vote
0
answers
71
views
What would go wrong if quantum observables were not represented by linear operators? [closed]
If quantum mechanical operators corresponding to physical observables were not hermitian, the corresponding eigenvalues may not be real. Since the eigenvalues are the outcomes of measurements of ...
- 213k
2
votes
1
answer
133
views
A coupled nonlinear dynamical system in four dimensional phase space
I have come across a coupled nonlinear dynamical system given below
$$ r\, \ddot{x} + \dot{x} = \sin y~,$$
$$ r\, \ddot{y} + \dot{y} = \sin x~,$$
where $r$ is some real number and $\dot{x}$ denotes $\...
- 213k
1
vote
0
answers
193
views
Why must operators in QM be linear?
Why must all operators in QM be linear (and therefore able to be represented by matrices). What is the physical reasoning behind this?
Is it be possible that the non-unitary nature of quantum collapse ...
- 213k
0
votes
0
answers
99
views
How an applied magnetic field breaks the inversion symmetry in a centrosymmetric system?
I want to understand why magnetic dipole transition breaks the inversion symmetry in a centrosymmetric system and gives rise to second-order nonlinearity.
- 1
10
votes
2
answers
682
views
Is LIGO able to detect the memory effect of gravitational waves?
Recently some articles in popular media1, 2 informed that LIGO will be able to measure the memory effect of gravitational waves described by Demetrios Christodoulou in 1991.3
The measurement method ...
- 2,666
1
vote
0
answers
76
views
Born-Infeld equation with a coefficient: which phenomena it describes?
Let us consider the well known Born-Infeld equation
$$-{\rm div}\left(\frac{\nabla u}{\sqrt{1-\frac{1}{b^2}|\nabla u|^2}}\right) =g(u).$$
It appears quite naturally in several fields such as ...
- 16.3k
0
votes
0
answers
257
views
How to find the steady state response of a Multi-Degree of Freedom (MDOF) system?
The Problem
I currently have a Multi-Degree of Freedom (MDOF) system with the following equation:
$$\mathbf{M\ddot{X}}+ \mathbf{D}(t)\mathbf{\dot{X}}^2 + \mathbf{C\dot{X}} + \mathbf{KX} = \mathbf{F}(t)...
- 213k
1
vote
1
answer
52
views
Linearization of 1D maps about a fixed unstable point [closed]
Recently, I was going through the paper Controlling Chemical Chaos in a three variable autocatalator system, by Peng et al. Here are the references
Although I have been introduced to 1D maps and the ...
- 12.7k
0
votes
2
answers
643
views
How do I non-dimensionalize Newton's Law of Gravitation for the 3-body problem?
I'm attempting to numerically solve the 3-body problem. Using Newton's second law, I've derived a system of 6 second order differential equations, the first three being:
$$ m_1\frac{d^2x_1}{dt^2} = -G ...
- 12.8k
1
vote
0
answers
48
views
How to generate a PDE from a discrete equation in a rice-pile like model?
I am reading Noise and dynamics of self-organized critical phenomena by Albert Díaz-Guilera Here, on an extension of the rice-pile model by Bak et al demonstrating self-organized criticality.
Equation ...
- 1,733
0
votes
1
answer
517
views
Definition of "nonlinear" in the context of perturbation of gravity
What exactly is the definition of a nonlinear perturbation when applied to a background spacetime metric?
I have seen so called "linear perturbations" which look like
$$ds^2 = -(1+2\Phi)dt^2 ...
- 213k
0
votes
1
answer
145
views
What the response of unstable limit cycles look like?
Stable limit cycles generate oscillations, i was wondering what the unstable limit cycles behaviours look like? From the picture in the left, the system shows a stable limit cycle and it generates ...
- 12.7k
1
vote
0
answers
36
views
Chaotic and Ordered Random Boolean Newtorks with a fixed in-degree k and a probability p
I'm working with Random Boolean Networks, I made a python program to show the dynamics of the networks. Before coding the program I study the theory and it says that the in-degree k and the ...
0
votes
1
answer
80
views
Higher order nonlinear stress definition
For the nonlinear case, I often find the following definition for the mechanical stress:
$$ \sigma=E_2\epsilon+E_3\epsilon^2$$
The parameters $E_2$ and $E_3$ are called "elastic modulus" or &...
- 28.1k
4
votes
1
answer
292
views
How chaotic is the double-pendulum if the arms are not perfectly rigid?
The double pendulum is a famous example of a chaotic system. It consists of one pendulum hanging from the end of another pendulum, which in turn hangs from a fixed point. In the traditional version, ...
- 12.7k
0
votes
1
answer
103
views
Derivation of dynamic nonlinear equation of motion of cantilever beam
Is it possible to derive the EOM of an inextensible cantilever beam without using any kind of variational principle I mean is it possible to derive it from Newton's law only?
Note: of course This ...
- 1,696
1
vote
3
answers
153
views
What's wrong with this information theoretic argument for free will? [closed]
Forgive me if this is incredibly naive. I am an undergraduate studying mathematics and have studied almost no physics, but a friend of mine mentioned this argument and it's been bugging me since it ...
- 12.7k
72
votes
7
answers
11k
views
Does gravity bend gravity?
Let's say that there is a large mass $M$ a light-year or so away from a black hole merger, which causes a very large gravitational wave to be produced. When the gravitational wave reaches $M$, does it ...
- 2,042
2
votes
0
answers
75
views
Nonlinear superposition and self-interaction in classical field theory [duplicate]
I am learning QFT (in a path integral formalism) and one thing I'm struggling with is that self-interaction is supposed to be a quantum phenomenon, not apparent in classical non-linear field theory. I ...
- 213k
2
votes
2
answers
57
views
Name for critical lines in parameter space and plots thereof
Suppose I study a dynamical system as a function of some control parameters, and I find that the nature of the attractors changes discontinuously (or non-analytically) at certain critical values (or ...
- 6,379
2
votes
1
answer
154
views
What type of bifurcation is this?
Consider the dynamical system
$$
dx/dt = -\cos(r)\sin(x)
$$
Clearly $x=0$ and $x=\pi$ are two fixed points of this system.
The stability of these two fixed points change as r is varied. Starting from $...
3
votes
2
answers
315
views
Stochastic system vs. stochastic process
I work on a project on stochastic diffusive systems described by stochastic differential equations (SDEs).
My background is from dynamical systems, so I tend to call the system under consideration a ...
- 65k
0
votes
2
answers
691
views
How to understand non-uniqueness of solutions of the Navier-Stokes Equations?
In the book of boundary layer theory:
"The solutions of the Navier–Stokes equations do not have to be unique for given initial and boundary conditions. Primarily because of the nonlinearity of ...
- 703
3
votes
1
answer
167
views
How to tell if a system has a direct or reverse energy cascade?
We know, in 3D turbulence one observes a direct energy cascade, where the energy flows from the large scales to small scales (see wiki 1,1), usually attributed to vortex stretching. We also know that ...
- 12.7k
0
votes
1
answer
72
views
Stability analysis of ODEs containing non-linear terms
I am currently reading this lecture notes on non-linear dynamics. If you look at equation (7) it is easy to write the ODEs,
$\dot{x} = y$ and $\dot{y} = -x$ into a matrix form $\dot{\vec{x}}=A\vec{x}$...
- 213k
1
vote
1
answer
94
views
Phase plots: The exact particular solution is a function of time, can't find fixed points. Now, in this situation, how to draw phase plots?
I want to draw phase plots.
The differential equations are two coupled second-order non-linear differential equations.
I have the exact particular analytic solutions.
However, the solutions are a ...
CommunityBot
- 1
1
vote
1
answer
126
views
Linear stability analysis of a 2-cycle
In a discrete $N$-dimensional Hamiltonian map $\mathbf{X}^{(n+1)}=f(\mathbf{X}^{(n)})$, we often find a 2-cycle which shows oscillation between two points in phase space. In such a Hamiltonian map we ...
- 4,496
2
votes
2
answers
155
views
What are the implications of deterministic chaos: useful or detrimental? [closed]
I am new to the concept of chaos theory and as a layman I am struggling to understand what is the significance and implication of chaos in ecological systems such as the chaotic predator prey model. I ...
- 12.7k
0
votes
0
answers
33
views
Fixed point of non-linear system: infinite eigenvector
I've come across a $2d$ non-linear dynamical system (autonomous) the stability properties of which I would like to understand better. Computing the stability matrix, its eigenvalues and eigenvectors, ...
- 213k
0
votes
3
answers
141
views
Solving non-linear function [closed]
I'm trying to solve the following function:
$$
R_c(x) = 100(x_2 − x_1^2)^2 + (1 − x_1)^2
$$
I need to find values of $x_1$ and $x_2$ as the value of $R_c$ changes. For example one solution to $R_c$ ...
- 4,496
0
votes
0
answers
48
views
Do you know about any book which discusses solitons in Benjamin-Ono Equation?
Benjamin-Ono equation is an integrable equation with soliton solutions. There are many books on solitons. The ones I know about mainly discuss solitons in Korteweg de-Vries and related equations. Do ...
- 2,666
0
votes
0
answers
65
views
Water vapor carbon dioxide temperature (warming) nonlinear feedback mechanism
In this Youtube video clip starting at 1:12:45 of the debate on the feedback mechanism of the CO$_2$-water-vapour warming effect Professor Richard Lindzen stated that the feedback mechanism is not ...
- 213k
2
votes
0
answers
38
views
How to perform the multi-scale analysis beyond harmonic oscillations?
I occasionally see this interesting method called multi-scale analysis. From what I understood, it is used to perturbatively solve a perturbed harmonic oscillator, meaning that the equation of motion ...
- 3,731
3
votes
1
answer
337
views
How does the linearity of the Schroedinger equation reflect the interactions?
There is a common lore that linear equations describe non-interacting systems, why non-linearities correspond to non-trivial interactions. My (loose) question is how is that compatible with the ...
- 55.3k
3
votes
2
answers
142
views
Uncertainty propagation in dynamical systems
I'm not a physicist, my training is in math and CS. If anything in this question is ill defined or doesn't make sense, say so in the comments and Ill try to fix it.
Suppose I have a discrete dynamical ...
- 12.7k
5
votes
1
answer
657
views
Non-linearity and self-coupling of gravity
I have heard that non-linearity of Einstein's field equations has to do with the fact that gravity self-couples. What does non-linearity have to do with self-coupling?
- 213k
1
vote
0
answers
44
views
Electromagnetism, linearity and Feynman diagrams [duplicate]
Currently I am reading Sean Carrol, general relativity. But a thing got me stuck in, I can't understand what he is talking about.
We are discussing the introduction to Einstein field equation, so he ...
- 65k
0
votes
0
answers
25
views
How to measure quantity difference between a nonlinear system of equations and its linearization?
I faced such a problem. I have a nonlinear system for control synthesis and I should compare not only my controllers but also a linear version of my system to describe the legitimacy of this ...
1
vote
0
answers
141
views
Solve nonlinear, forced and damped Duffing oscillator
I am trying to solve a Duffing type equation by using Van Der Paul's method:
\begin{align}
\ddot{x} + \omega^2 x + 2 \gamma \dot{x} + \beta x^3 = f \cos(\Omega t)
\end{align}
with $$x(t) = Re[A(t) \...
0
votes
0
answers
35
views
Kinetic description of an autocatalytic system
I am looking at the system:
$2X + Y \Leftrightarrow 3X $
$A \rightarrow Y$
$X \rightarrow B$
The description in terms of differential equations is this:
$\frac{dx}{dt} = x^{2}y - x$
$\frac{dy}{dt} = a ...
- 113
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 ...
- 213k
12
votes
2
answers
13k
views
Nonlinear spring $F=-kx^3$
A nonlinear spring whose restoring force is given by $F=-kx^3$ where $x$ is the displacement from equilibrium , is stretched a distance $A$. Attached to its end is a mass $m$. Calculate....(I can do ...
- 2,509
0
votes
1
answer
144
views
Differential equation for the anharmonic oscillator
In my project me and my partner used the engine to constrain the system so we can see the anharmonic oscillations. In our first analysis we get only odd powers in differential equation, so there ...
- 12.8k