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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Non-linear dynamics

I am confusing between chaotic attractor and strange attractor. I think strange attractor also is related to nonlinear dynamics. Then what is the difference between chaotic attractor and strange ...
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Questions about Taylor Series Expansion of an Effective Potential [closed]

I was reading a paper and I had a question about it. Here is the relevant part of the paper Here is what I have understood so far: The authors start off with a hamiltonian which describes a particle ...
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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.
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Can this problem be treated in purely mathematical way other than the graphical way? [migrated]

Problem Statement: I attempted all the parts. For part (a), $E_1 = P + R + S$, so $ \dot E_1 = \dot P + \dot R + \dot S$. Plugging these values and rearranging the terms would get the desired ...
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Higher order nonlinear ultrasonic signals

With nonlinear ultrasound, the higher harmonic frequencies are used, for example, to identify defects in materials. Usually the second and third harmonic frequencies are used. If higher orders are ...
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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 ...
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Is this system linear? [migrated]

$$y(t)=\begin{cases}0 \hspace{4.86cm}x(t)<0\\ x(t)+x(t-2)\hspace{2cm} x(t)\geq0\end{cases}$$ Assuming $C>0$. If $x(t) = C$ is an input, the output will be $2C$, and the output of $-x(t)=-C$ must ...
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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)...
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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 ...
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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 ...
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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 ...
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Parametric Resonance Analysis using Perturbative approach

I'm reading Parametric Resonance from Landau's Mechanics Text. A similar calculation is done here. Supposing a parametric oscillator given by $$\ddot{x}(t)+\omega_0^2(1+h\cos(\gamma t))x(t)=0$$ It's ...
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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 ...
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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 ...
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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 &...
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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, ...
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Particle and metric redefine in action corresponding and beyond conformal transformation

let us beginning form action with system of gravity and scalar field. $$ S=\int d^d x \sqrt{g}(R-g^{ab}\partial_a \phi \partial_b \phi-V {(\phi)} )$$ and redefine metric $$ \tilde{g_{ab}} = f(g_{ab},\...
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Do these Lagrange equations of 1st kind exhibit numerical instabilities?

I followed the lead of "Theoretische Physik", 1e, 2015 by Bartelmann et al. (pp. 171 - 174) to form the set of constituting Lagrange equations of the 1st kind for the double pendulum: eight ...
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RG flow diagram plotting

I want to be able to plot a flow diagram with a given recursion relation. For example, I have the follow recursion relation: \begin{align*} \frac{dT}{d\ell} &= 2T{y_0}^2 a^2 \\ \frac{d ...
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1answer
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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 ...
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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 ...
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How does Born-Infeld nonlinear electrodynamics classically explain QED vacuum polarization?

The Born-Infeld model of nonlinear electrodynamics is described by the following Lagrangian $^1$ $${\cal {L_{\rm {BI}}}} = 4b^2\left( 1-\sqrt {1+\frac {F}{2b^2} } \right),$$ where $b$ is a maximum ...
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Radiative corrections to Coulomb’s law and Euler-Heisenberg theory

Maxwell's electrodynamics is the classical limit of QED (quantum electrodynamics). Using Maxwell's equations, the electrostatic (Coulomb) potential of a point charge is obtained as $\Phi \propto \frac{...
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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 ...
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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 ...
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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 ...
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2answers
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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 ...
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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 $...
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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 ...
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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 ...
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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 $\dot{\vec{x}}=A\vec{x}$...
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Generating nonlinearities in renormalization group

In renormalization group (RG) calculations as performed in statistical physics (for example for Landau-Ginzburg theory - often a la Wilson), the first step is to coarse-grain the theory by integrating ...
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Relative Phase distribution for two coupled van der Pol oscillators where one has a Drive?

I was solving the dynamics for a driven coupled (inertial) van der Pol oscillators, where only one oscillator is driven. I started with the complex amplitudes $\alpha$ for both of the systems which ...
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1answer
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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 ...
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1answer
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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 ...
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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, ...
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2answers
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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 ...
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3answers
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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 $R_c$ ...
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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 ...
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Solving coupled propogation equations for EM waves

I have recently come across a set of partial differentials that describe the propogation of two coupled EM fields, in a 2D system currently being investigated. In their most general form they are $\...
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Stability analysis of a cubic characteristic polynomial

I have the following cubic characteristic polynomial describing some dynamical system: $$f(\lambda, b) = \lambda^{3} + 3\left(\frac{1}{2} + i \right)\lambda^{2} + \left( 3i-3b-\frac{1}{4} \right)\...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) \...
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Generalizations of Feigenbaum universality for multidimensional maps and ones with multiple order parameters

Feigenbaum showed that for discrete 1D dynamical systems with a (smooth) unimodal evolution function, the route to chaos is universal, and depends only on the order of the map's maximum. (I'm told ...
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Dynamical system in two dimensions

I am trying to expand system into two dimensions. Dynamical system with damping proportional to squared velocity is given by Newton's eq: $$ \ddot{x}(t) + \mu \dot{x}^2(t) = 0.$$ Now, I want my system ...

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