Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [non-linear-systems]

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.

1
vote
0answers
57 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?
0
votes
0answers
19 views

Spontaneous synchronization references

Can someone suggest references for an introduction on spontaneous synchronization, theory/examples. I am trying to understand it so I can test it for some problems I am working on. I have no prior ...
1
vote
0answers
22 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 $u$. ...
0
votes
1answer
36 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: ...
0
votes
0answers
17 views

About FPU non linear problem, in reference to the original article

I'm reading the original article about the Fermi-Pasta-Ulam-Tsingou (FPUT) problem and I have some problems about the conclusion. Here the behavior of the system as was reported in the article: $$x_i=(...
2
votes
1answer
74 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?
0
votes
0answers
22 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 ...
0
votes
0answers
45 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, ...
0
votes
1answer
18 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 $ \...
0
votes
0answers
39 views

Lagrangian for non-smooth mechanical systems (bouncing ball)

I've been searching for a while for this answer, but yet not found anything. Let's take a bouncing ball as an example, what would the Lagrangian be of this system? In this case we can use the ...
0
votes
1answer
88 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 ...
-1
votes
3answers
67 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}...
0
votes
0answers
55 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 ...
0
votes
1answer
16 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, ...
2
votes
0answers
253 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
votes
1answer
47 views

Poincaré plane and Logistic Map

How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
2
votes
2answers
113 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) = \...
4
votes
2answers
86 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 ...
3
votes
1answer
68 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 ...
2
votes
1answer
112 views

Is *every* planar/2D system integrable?

Consider the generic following 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 by ...
0
votes
1answer
67 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 ...
0
votes
2answers
61 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 ...
0
votes
0answers
122 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 ...
1
vote
2answers
63 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 been used to describe stable points for periodic motion. I have also read about how bifurcation diagrams describe changes in ...
1
vote
1answer
104 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?
2
votes
2answers
57 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? ...
2
votes
1answer
67 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 ...
0
votes
1answer
49 views

Defining a 'small disturbance which dampens in time' while identifying stable points in a nonlinear system

I'm reading the book "Nonlinear dynamics and Chaos" by S Strogatz. In section 2.2, titled "Fixed points and stability", he defines equilibrium points as solutions where ...all sufficiently small ...
1
vote
1answer
32 views

Reynolds decomposition of non-linear dynamics

Can we apply the Reynolds decomposition, $$u(x,t)=U(x)+u'(x,t),$$ to any strongly non-linear dynamics problem, where the final state is dependent on the initial condition like Lorenz's equation or ...
2
votes
3answers
148 views

Linear and non-linear systems

When I read about the superposition principle, it says that it works only on linear systems, my problem is that I cannot really understand the difference between a linear and a non-linear system. I ...
2
votes
2answers
54 views

Closed gravitational orbits and gradient systems

I am currently studying non-linear dynamics on my own time. One of the theorems in the material is that systems that can be written as gradient problems cannot have closed orbits i.e. systems like $$\...
1
vote
1answer
87 views

Non-abelian gauge theories are non-linear

To preface this, I know very little about Standard model physics and nonabelian gauge theory, so please correct me if my understanding is incorrect. I was reading about the Standard Model, and I ...
1
vote
1answer
28 views

Formation of patterns in instabilities according to the wavelength of instability

Do the wavelengths of the instability have an influence on the type of patterns that we are going to get? Meaning for example in 2D, if the unstable wavelengths are small with respect to the size of ...
1
vote
2answers
35 views

Is the trajectory of a particle with constant velocity (though its direction can change by collisions) always non-chaotic?

Suppose we have a particle that travels with constant velocity, without heat losses by friction, and no forces acting on it except for occasionally collisions with much bigger wall-like masses than ...
2
votes
1answer
50 views

Spring non-linear behavior for small forces

While writing a report about a classical spring experiment I noticed that, if small forces were applied to our spring, this would stretch much less than expected. Searching on the internet I've kinda ...
4
votes
3answers
108 views

Link between integrability and soliton solutions

I have been doing some research on the properties and dynamics of solitons (in particular, solitons in superfluids) and several works and papers mention the link between solitonic solutions and ...
2
votes
1answer
97 views

In what physical situations is the weak-field limit invalid?

in the weak-field limit gravitation is described by a symmetric tensor field $h_{μν}(x)$ in flat spacetime. Linear theory suffices for nearly all experimental applications of general relativity ...
2
votes
0answers
38 views

How to use Belinsky-Zakharov transformation

I know it might be trivial. When using BZ transformation [1] to generate soliton solutions of Einstein’s field equations, one need a seed solution $g_{0}$ which gives $A_{0}$ and $B_{0}$. Taking them ...
3
votes
2answers
116 views

Chaos implies Nonlinearity?

Why, for finite dimensions, is nonlinearity a precondition for chaos? This article (Linear Chaos? By Nathan S. Feldman) offers an example of an infinite dimensional chaotic map, which is linear. ...
0
votes
0answers
221 views

Maxwell's equations and nonlinear media

Are there analytical methods to analyze electromagnetic fields or magnetic diffusion in materials which are not linear using (or starting from) Maxwell’s equations? Nonlinear material could be ...
1
vote
1answer
80 views

Classification of fixed points in 4D phase space

The usual classification of fixed points as used in linear stability analysis is based on planar systems (un-/stable node, un-/stable spiral point, saddle). I need to extend this classification to a ...
1
vote
1answer
160 views

Nonlinear dynamics and chaos theory in electrical systems and circuits [closed]

Is there a possible application of the field of chaos theory and nonlinear dynamics to electrical systems such as circuits and power? If so, are they based on conventional nonlinear dynamics or are ...
0
votes
1answer
57 views

Book on Non-Linear dynamics [duplicate]

I am currently planning on self studying Non-linear dynamics, with an intention of developing a decent idea about it and see how its applied. I don't want to be too rigorous in my dealing with the ...
1
vote
1answer
79 views

Question about Strange Attractors

I'm reading the book Chaos by James Gleick and came upon a certain excerpt in the chapter 'Strange Attractors'. I'm having a hard time understanding it (the merging of two surfaces part, in particular)...
0
votes
1answer
74 views

Understanding Boyle's Law and Charles's Law

Boyle's Law is defined as follows: $PV=k$ This implies that $P_{1}V_{1}=P_{2}V_{2}$ is true while temperature and mass of confined gas is constant. This would mean that $P_{2}=P_{1}V_{1}/V_{2}$ ...
3
votes
1answer
144 views

Free energy in Allen-Cahn PDE

I am a mathematician and I am taking a mathematical physics course. In the part of reaction-diffusion equations, there is something that I do not understand. I have been defined the Allen-Cahn ...
2
votes
1answer
88 views

Why some dynamic systems can undergo sudden changes?

Everybody has observed that the weather may change from beautiful sunshine to extremely bad weather (heavy rain, stormy winds, ...) within less than half hour. What is the fundamental reason for this? ...
-1
votes
1answer
280 views

Is an 'angle of slope' really the same on Earth and Moon? [closed]

I know (and it's easy to proof the formula), that the maximum angle at which an object will stay static on a slope being at an $\alpha$ angle to the ground is $$\tan \alpha = \mu$$ where $\mu$ is the ...
1
vote
0answers
46 views

Nonlinearities arising from linear equations

I have sources that tell me that the Bloch Equations, which describe the magnetization vector in nuclear magnetic resonance, are nonlinear. In vector form, without relaxation, they are: $$ \partial_t \...
3
votes
1answer
216 views

What is a diabolical point?

A lot of papers define a 'diabolical point' as a "double semi-simple eigenvalue." I know a semi-simple eigenvalue is one which has algebraic multiplicity and geometric multiplicity to be equal. ...