Questions tagged [bifurcation]

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Energy delivered to unstable limit cycle

For a given family of stable and unstable $T$-periodic limit cycles $\Gamma$ forming a manifold $\mathcal{M}\subset \mathbb{R}^n\times \mathbb{R}^p$ of some (nonconservative) $p$-parametric $n$-...
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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 ...
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Inhomogeneous Brusselator model

I am studying a Brusselator model which includes inhomogeneous terms such as $\nabla ^2 x$ and $\nabla^2 y$ in the following way \begin{aligned} &\partial_{\tau} x=a-(b+1) x+x^{2} y+d_{X} \nabla^{...
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Graph Interpretation of Fold of Cyclic Bifurcation

I was studying about local bifurcations of cycles, and the basic type of it was the cyclic fold. According to the textbook, cyclic fold bifucation was the meaning of one of those that is, for two ...
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Terminology for scenario when energy of system $E(\theta_1,\ldots,\theta_k)$ with $k$ real parameters, is minimum whenever $\theta_1=c$ (fixed value)

Disclaimer. I'm not a physicist. Consider a physical system whose "energy" $E$ is a function of $k$ real parameters $\theta = (\theta_1,\ldots,\theta_k) \in \mathbb R^k$. Let $E_{\min}$ be ...
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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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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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Solutions to Chen System/Attractors

I have a problem involving the new chaotic system dubbed as the Chen System. This involves a system of coupled nonlinear ordinary differential equations. My problem is to determine for which ...
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Malkus Lorenz water wheel

I have read a paper by Leslie E. Matson named "The Malkus-Lorenz water wheel revisited" published in the American Journal of Physics in 2007. There the author has discussed about the ...
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How can I use bifurcation analysis of the Lorenz system in calculating the fractal dimension by the Spectral decay coefficient method?

Discrete Fourier transform represents data by a superposition of sines and cosines that have various amplitudes and frequencies. With time series of length N, the range of frequencies that can be ...
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Bifurcation of Van Der Waal equation for real gases

Van Der Waal's equation leads to a cubic equation in v of the form $$Pv^3-(bP+RT)v^2+av-ab=0$$ This equation has 3 roots for $T<T_C$ and one root for $T>T_C$ I understand why region ABCDE is ...
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Bifurcations in Statistical Physics

I am currently a grad level student in physics with much interest in statistical and soft-matter physics (equilibrium and out of equilibrium); I am currently taking a course in numerical methods for ...
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Path between fixed points in logistic map

I have a question about period doubling and fixed points in the logistic map. Let's say I have a basic logistic map, $$f(x) = 4\lambda x(1-x).$$ Let me then compare 1,2 and 4 iterations of this map on ...
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Superfluid Vortex Lines Bifurcation and Winding Numbers

Vortex lines in superfluids are characterized by their quantised circulation: $k = \frac{h}{m}\times n$, where $n$ is the winding number in the sense of a topological winding number. Now, most vortex ...
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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 ...
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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) ...
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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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Issue with Bifurcation Plot for Driven Pendulum

I'm trying to create a bifurcation plot for a driven damped pendulum. In particular, I'm trying to recreate the plot found in Taylor's 'Classical Mechanics' (page 484) for a driving strength $\gamma$ ...
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1 answer
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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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What causes the emergence of patterns in a fluid (the " in between" flows are chaotic) in rotating Couette cells?

In this video, around 26 minutes and 30 seconds, you can see that a fluid (whose velocity is made visible) in a Couette cell shows nice patterns at certain rotation velocities, while between these ...
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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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What is linear / eigenvalue buckling analysis?

I need some simple and clear explanation of what is called linear buckling analysis and why it is also called eigenvalue buckling analysis? In other words how natural vibration frequency or ...
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Non-Linear Behavior of Iterated Functional Maps

The universal behavior of certain iterated nonlinear function maps (ie period doubling bifurcation route to chaos): $$x_{i+1}=f(x_i)$$ have been known since Feigenbaum: (see http://...
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Solutions of symmetric equations are not invariant - is symmetry spontaneously broken?

I have a system of equations: \begin{cases} f\left(x_{1}\right)+f\left(x_{2}\right)+P=0\\ \\ g\left(x_{1}\right)+g\left(x_{2}\right)=0 \end{cases} where $f$ and $g$ are some functions, $P$ is a ...
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What can I expect to see in a oscillator exhibiting bifurcation?

I have a program which aims to simulate a Josephson Bifurcation Amplifier. I am currently trying to obtain a plot of the probability of bifurcation as a function of the ratio between the driving and ...
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