All Questions
Tagged with complex-systems oscillators
17 questions
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Lyapunov Exponent for Double Pendulum
I want to calculate the Lyapunov Exponent for a double pendulum, with a small change in the initial angle. In this study, the authors used the formula $\frac{1}{t}{ln(\frac{d}{d_0})}$ as $t$ tends to ...
- 136
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1
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88
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Are there any "linear" lagrangian systems of interest for which an analytic solution is not obvious?
Out of curiosity, I am interested in Lagrangian dynamical systems that can be expressed in a "linear" manner. By this, I mean that their Lagrangian can be expressed, quadratically, as
$$L = \...
- 175
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3
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613
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How to justify this small angle approximation $\dot{\theta}^2=0$?
Suppose the equation of motion for some oscillating system takes the following form:
$$\ddot{\theta}+\dot{\theta}^2\sin\theta+k^2\theta\cos\theta=0$$
Applying small angle approximation to $\theta$ ...
- 205
3
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1
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91
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What is the simplest PDE/ODE/model I can use to understand how nonlinearities can lead to leakage of energy to higher harmonics in an oscillator?
I came across this problem in the study of surface waves in an oscillating cylindrical vessel of liquid.
There are various eigenmodes described using Bessel functions, and energy transfer can happen ...
- 71
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1k
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A rule for when phase-space orbits may cross
Note: in this question when I talk about "phase space," I will be refering to velocity vs. position space, which can also be correctly referred to as "state space." Many sources (...
- 1,394
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51
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How to get the limit cycle of a nonlinear oscillator using Shooting Method?
I am trying to find the limit cycle of a nonlinear oscillator like a duffing or Van-der-Pol oscillator directly using the shooting method. I know how to use the Shooting method to solve a Boundary ...
- 2,235
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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$-...
- 101
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Self-synchronizing and -desynchronizing systems of oscillators
There are biological systems with adaptable frequencies that are able to synchronize their frequencies, mainly individuals (see e.g. reproductive synchrony). In this case, also the phase is typically ...
3
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4
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250
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Asymptotic frequency of nonlinear oscillator $\ddot x = -x-{\dot x}^3$ (speed cubed)
A particle oscillates according to the equation
$\ddot x = -x-{\dot x}^3.$ The positive positions of the particle when it changes direction, $\dot x = 0$, are $x_1,x_2,\ldots$.
I want to show that
$$\...
- 131
2
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1
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782
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How do i read a pendulum phase diagram?
I'm trying to understand how the intermediate axis theorem works. And in one of the works that I found, they used a pendulum phase diagram, but idk how to read it. Can anybody help please?
The work ...
- 338
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236
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Are all orbits of the conservative pendulum homoclinic?
I don't understand this statement:
"The homoclinic orbit is characterized by $E = mgl$. When $E < mgl$, the pendulum is tracing other orbits."
If energy is conserved, then $E_0 = E$ ($E$...
- 202
3
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1
answer
261
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Is there a relation between large-scale oscillations and small-scale oscillations?
From Neural oscillation - Wikipedia:
Oscillatory activity in the brain is widely observed at different levels of organization and is thought to play a key role in processing neural information.
In ...
- 929
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2
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1k
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Significance of the word 'linear' in linear harmonic oscillator
In my book Advanced Acoustics there is a line-
A particle undergoing SHM is called a linear harmonic oscillator
If I say that the word linear is used for the 2 reasons-
The motion of the particle ...
- 83
1
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2
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161
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Is there a relation between phase plane and complex plane?
The only occurrence I see complex numbers used in dynamical systems is to analyse the eigenvalue $\lambda$ of the linearised approximation to determine the characteristics of equilibrium points. ...
- 929
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263
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What branch of physics covers most of these questions?
I am close to finish the book Vibration and Wave by French, and I would like to know which branches of physics can answer these groups of questions:
Defining questions: Can a periodic event be ...
17
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13k
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What creates the chaotic motion on a double pendulum?
As we know, The double pendulum has a chaotic motion. But, why is this? I mean, the mass of the two pendulums are the same and they have the same length. But, what makes its motion random?
I'm just ...
- 1,107
2
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2k
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Linearized equations
What is $V_{\alpha\beta}$?
And what is a symmetric, positive definite potential energy matrix?
And why is there a linearized equation like this?
- 131