Questions tagged [bifurcation]

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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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55 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 ...
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21 views

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
19 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 ...
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39 views

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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41 views

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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26 views

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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21 views

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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46 views

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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55 views

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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1answer
59 views

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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40 views

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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1answer
44 views

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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1answer
89 views

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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2answers
95 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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2answers
699 views

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 $\gamma$ ...
2
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1answer
142 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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1answer
41 views

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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1answer
606 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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1answer
2k views

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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108 views

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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2answers
150 views

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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172 views

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 ...