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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.

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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 $ \...
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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 ...
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80 views

Solution of the coupled non-linear oscillators by using perturbation theory [on hold]

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 ...
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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}...
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54 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 ...
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1answer
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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, ...
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248 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 ...
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38 views

Poincaré plane and Logistic Map

How can we draw Poincaré plane and phase portrait for the Logistic Map for different parameter values?
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106 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) = \...
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2answers
72 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 ...
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53 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 ...
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101 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 ...
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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 ...
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2answers
56 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 ...
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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 ...
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Susceptibility in Bifurcations

In dynamical systems a bifurcation describes the phenomenon where, as a system parameter is varies, a qualitative property of the system might be altered. Similarly, in statistical physics, a ...
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57 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 ...
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Non-Linear Sigma Model

I read C. Mudry's book (chapter 3, p. 72, eq. 3.1a (the definition) and eq. 3.2a) and have several quaestions (1) He defines NL$\sigma$M through the following partition function: $$\mathcal{Z}(N,\...
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97 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?
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50 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? ...
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1answer
63 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 ...
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46 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 ...
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31 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 ...
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3answers
141 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 ...
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46 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 $$\...
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71 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 ...
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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 ...
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2answers
34 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 ...
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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 ...
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3answers
101 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 ...
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1answer
91 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 ...
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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 ...
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107 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. ...
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What does it mean by 'unfolding of a pitchfork bifurcation'?

I am analyzing a dynamical system, where due to a small imperfection the original perfect bifurcation structure gets disturbed and leads to complex emerging bifurcations, one of which is a ...
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1answer
69 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 ...
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1answer
138 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 ...
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1answer
52 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 ...
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1answer
75 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)...
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68 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 $P_{2}=P_{1}V_{1}/V_{2}$ ...
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133 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 ...
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1answer
85 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? ...
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248 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 ...
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44 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 \...
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167 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. ...
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1answer
25 views

When can raw data can be used to look for phase synchronization between time series?

I'm studying a system formed by multiple rotors non-linearly coupled to each other. If I want to look for phase synchronization, I could just look at the angle for predefined points around the ...
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3answers
297 views

Non-linear dynamics problem: A mechanical analog of dx/dt=sinx [closed]

I have been stuck at this particular problem for a while.This is a problem from Nonlinear Dynamics And Chaos by Strogatz. The thing I am having hard time finding a mechanical system following dx/dt=...
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1answer
228 views

Are bifurcations in dynamical systems related to phase transitions? [closed]

Bifurcation is a qualitative measure for a dynamical system changing the system parameter. Does the statistical behavior in the system shows phase transition-like characteristics?
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81 views

Are there some necessary and sufficient conditions that a system can be modeled using Monte Carlo Methods?

Is there a certain set of conditions a system need to follow such that it can be effectively modeled using any "general" Monte Carlo method, maybe to find some average value of a thermodynamic ...
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84 views

How do nonlinear phenomena arise from linear theories

How is it possible that linear theories, for example maxwells equations or the schroedinger equation, produce nonlinear physics?
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147 views

Part1 — Beginner level confusion regarding terminologies — symbolic dynamics, trajectory, phase space

I came across the topic of symbolic dynamics when studying about time series analysis. Since I have not formally taken any course on chaotic dynamics, I have some difficulties in understanding some ...