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In this paper, a new triangular decomposition algorithm is proposed for ordinary differential polynomial systems, which has triple exponential computational complexity. The key idea is to eliminate one algebraic variable from a set of…
In this paper the Benettin-Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo's derivative and the corresponding Matlab code are presented. First it is proved that the considered…
Lyapunov exponents of a dynamical system are a useful tool to gauge the stability and complexity of the system. This paper offers a definition of Lyapunov exponents for a sequence of free linear operators. The definition is based on the…
We develop a practical approach to establish the stability, that is, the recurrence in a given set, of a large class of controlled Markov chains. These processes arise in various areas of applied science and encompass important numerical…
We study the Lyapunov exponents $\Lambda(x)$ for Markov dynamics as a function of path determined by $x\in \mathbb RP^1$ on a binary planar tree, describing the Markov triples and their "tropical" version - Euclid triples. We show that the…
A recently developed method for the calculation of Lyapunov exponents of dynamical systems is described. The method is applicable whenever the linearized dynamics is Hamiltonian. By utilizing the exponential representation of symplectic…
We consider $2p\ge 4$ order differential operator on the real line with a periodic coefficients. The spectrum of this operator is absolutely continuous and is a union of spectral bands separated by gaps. We define the Lyapunov function,…
Working on strongly irreducible planar self-affine sets satisfying the strong open set condition, we calculate the Birkhoff spectrum of continuous potentials and the Lyapunov spectrum.
We compare three methods for computing invariant Lyapunov exponents (LEs) in general relativity. They involve the geodesic deviation vector technique (M1), the two-nearby-orbits method with projection operations and with coordinate time as…
We consider periodic matrix-valued Jacobi operators. The spectrum of this operator is absolutely continuous and consists of intervals separated by gaps. We define the Lyapunov function, which is analytic on an associated Riemann surface. On…
The Discrete Morse Theory of Forman appeared to be useful for providing filtration-preserving reductions of complexes in the study of persistent homology. So far, the algorithms computing discrete Morse matchings have only been used for…
We discuss several techniques for the evaluation of the generalised Lyapunov exponents which characterise the growth of products of random matrices in the large-deviation regime. A Monte Carlo algorithm that performs importance sampling…
We study the dimension spectrum for Lyapunov exponents for rational maps acting on the Riemann sphere and characterize it by means of the Legendre-Fenchel transform of the hidden variational pressure. This pressure is defined by means of…
We study the spectral transition line of the extended Harper's model in the positive Lyapunov exponent regime. We show that both pure point spectrum and purely singular continuous spectrum occur for dense subsets of frequencies on the…
This paper addresses the stabilization issue for fractional order switching systems. Common Lyapunov method is generalized for fractional order systems and frequency domain stability equivalent to this method is proposed to prove the…
We prove non-trivial upper and lower bounds for the "Spectrum of Singularities" of Fourier Series with polynomial frequencies. The Spectrum of Singularities of a function f gives the Hausdorff dimension of the set of points with a given…
We describe all Lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism. The description is expressed in terms of the finest dominated splitting and Lyapunov exponents that appear in the…
This paper is devoted to study multifractal analysis of quotients of Birkhoff averages for countable Markov maps. We prove a variational principle for the Hausdorff dimension of the level sets. Under certain assumptions we are able to show…
In this paper we use a path-integral approach to represent the Lyapunov exponents of both deterministic and stochastic dynamical systems. In both cases the relevant correlation functions are obtained from a (one-dimensional) supersymmetric…
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…