Questions tagged [lyapunov-exponents]
Lyapunov exponents (not to be confused with Lyapunov functions) are gadgets that describe the exponential rate at which the trajectories of infinitesimally close initial conditions diverge from one another under a certain time evolution. They were first considered in the context of the qualitative theory of ODE's; now they are used in a variety of disciplines; in particular they are fundamental objects in smooth ergodic theory.
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In how far is the Lyapunov spectrum characteristic for a dynamic system?
If two dynamic systems have the same Lyapunov spectrum on their respective attractor (of equal dimension, of course), which results relate to the properties of these two dynamic systems on those ...
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Can probe trajectories to compute Lyapunov exponents get "stuck in more regular orbits" after rescaling?
I am computing Lyapunov exponents, and there is something that I do not understand about the data.
The model has a regime for $\delta \approx 1$ (in some units) where it is fully chaotic, and the ...
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Computing Lyapunov Exponents of an Example of Avila and Bochi
In Artur Avila and Jairo Bochi's lecture notes (see here: http://mat.puc-rio.br/~jairo/docs/trieste.pdf) in section 3.1 they deal with Lyapunov exponents of products of random i.i.d. matrices. Let $\{...
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Top Lyapunov Exponent of a cocycle is invariant under the map.
Given an map invertible measurable $T$ on a space $(X,\mu)$, take a linear cocycle $\mathcal{A}(x,n)$.
The top Lyapunov exponent at $x$ is defined as
$$\chi(x)=\limsup_{n \to \infty}\frac{1}{n}\log{\|\...
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Does the Hamiltonian system have unbound solutions?
I want to know if it is possible to determine if the following Hamiltonian system has unbound solutions. Let us consider the Hamiltonian function
$$ H(x,y,p_x,p_y) = \frac{1}{2}(p_x^2+p_y^2) + \frac{x^...
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Is this a viable way to calculate the Lyapunov Exponent?
Is this a viable way to compute the Lyapunov Exponent for a double pendulum?
Here is the code for the double pendulum (You don't have to look at this part, it just returns the angles and angular ...
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Relation between symmetries and Lyapunov exponents
Let us consider a system i) that is Hamiltonian, and ii) where we can apply the Oseledets theorem.
The presence of a symmetry ensures the presence of a vanishing (zero) Lyapunov exponent.
To be more ...
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Lyapunov methods to show the trajectories that do not converge to the origin [closed]
Consider the map $f(z) = z^2$, where $z$ represents complex numbers.
what is the function that $f$ corresponds to the map $p(r, \theta) = (r^2, \theta)$ in polar coordinates. And is it true that all ...
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Lyapunov Exponents for $n$-Dimensional Matrix $A$
I am wondering whether my solve is correct. I know how to solve the 2, or 3 dimension of the state matrix. But what if the state matrix goes to n-dim?
Here is what I tried:
To find the Lyapunov ...
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Action of the adjoint operator in tangent space
I'm reading a paper on the computation of covariant Lyapunov vectors (https://arxiv.org/pdf/1212.3961.pdf) and, as I have a Machine Learning background, I have some gaps concerning dynamical systems.
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Connection between ergodicity and Lyapunov exponents
This will be a soft reference question in a sense, as I will not state any rigorous theorems/results. Whenever I happen to read about ergodic systems, be it a specific book, article or a blog post, a ...
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Can Jacobi Fields be used to explicitly calculate Lyapunov exponents for geodesic flows?
My background is in geometry, and I have just become interested in the ergodic theory of geodesic flow. In Sarnak's 1981 paper
"Entropy estimates for geodesic flows," the following ...
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A Lemma Related to Oseledets' Theorem
Let $(X,\mu)$ be a probability space, $T:X\to X$ measurable and preserves $\mu$, $A:X\to GL_d(\mathbb{R})$ measurable and $\log^+\Vert A\Vert$, $\log^+\Vert A^{-1}\Vert$ integrable. Define $A^1(x)=A(x)...
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Why do we refer to Lyapunov exponents as a characteristic of the system, telling us about its chaoticity, when it is only referred to a point?
Lyapunov exponents are defined as indicators of the sensibility to initial conditions. In fact, they give the mean rate of exponential separation of trajectories. They are, however, specific of a ...
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About Lyapunov exponent. Can I say that a smaller Lyapunov exponent (negative) means "better" stability?
For a equilibrium position $x_0$ and a perturbed (or with error) one $x_0^\prime:= x_0 + \epsilon_0$. Their distance after time $t$ is $\epsilon_t \approx \epsilon_0 e^{t \lambda(x_0)}$. Can I say ...
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