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Guido Gentile - One of the best experts on this subject based on the ideXlab platform.
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Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies
arXiv: Dynamical Systems, 2020Co-Authors: Guido Gentile, Faenia VaiaAbstract:We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same Frequency Vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non resonance condition on the Frequency Vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors problem, we confine ourselves to two-dimensional Frequency Vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the Frequency Vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with "holes". If we want the set to be connected we have to require some non-resonance condition on the Frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.
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Response solutions for forced systems with large dissipation and arbitrary Frequency Vectors
Journal of Mathematical Physics, 2017Co-Authors: Guido Gentile, Faenia VaiaAbstract:We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is, quasi-periodic solutions having the same Frequency Vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the Frequency Vector. In this paper we show that the results extend to the case of arbitrary Frequency Vectors.We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is, quasi-periodic solutions having the same Frequency Vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the Frequency Vector. In this paper we show that the results extend to the case of arbitrary Frequency Vectors.
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Resonant tori of arbitrary codimension for quasi-periodically forced systems
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Livia Corsi, Guido GentileAbstract:We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility property and we assume its Frequency Vector to be Bryuno. Then we prove that, without imposing any non-degeneracy condition on the forcing, there exists at least one quasi-periodic solution with the same Frequency Vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases.
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convergent series for quasi periodically forced strongly dissipative systems
Communications in Contemporary Mathematics, 2014Co-Authors: Livia Corsi, Roberto Feola, Guido GentileAbstract:We study the ordinary differential equation eẍ + ẋ + eg(x) = ef(ωt), with f and g analytic and f quasi-periodic in t with Frequency Vector ω ∈ ℝd. We show that if there exists c0 ∈ ℝ such that g(c0) equals the average of f and the first non-zero derivative of g at c0 is of odd order 𝔫, then, for e small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c0, with the same Frequency Vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for 𝔫 = 1. We also point out that, if 𝔫 = 1 and the first derivative of g at c0 is positive, then the quasi-periodic solution is locally unique and attractive.
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Domains of analyticity for response solutions in strongly dissipative forced systems
Journal of Mathematical Physics, 2013Co-Authors: Livia Corsi, Roberto Feola, Guido GentileAbstract:We study the ordinary differential equation ɛx+x+ɛg(x)=ɛf(ωt), where g and f are real-analytic functions, with f quasi-periodic in t with Frequency Vector ω. If c0∈R is such that g(c0) equals the average of f and g′(c0) ≠ 0, under very mild assumptions on ω there exists a quasi-periodic solution close to c0 with Frequency Vector ω. We show that such a solution depends analytically on ɛ in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.
Faenia Vaia - One of the best experts on this subject based on the ideXlab platform.
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Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies
arXiv: Dynamical Systems, 2020Co-Authors: Guido Gentile, Faenia VaiaAbstract:We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same Frequency Vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non resonance condition on the Frequency Vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors problem, we confine ourselves to two-dimensional Frequency Vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the Frequency Vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with "holes". If we want the set to be connected we have to require some non-resonance condition on the Frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.
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Response solutions for forced systems with large dissipation and arbitrary Frequency Vectors
Journal of Mathematical Physics, 2017Co-Authors: Guido Gentile, Faenia VaiaAbstract:We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is, quasi-periodic solutions having the same Frequency Vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the Frequency Vector. In this paper we show that the results extend to the case of arbitrary Frequency Vectors.We study the behaviour of one-dimensional strongly dissipative systems subject to a quasi-periodic force. In particular we are interested in the existence of response solutions, that is, quasi-periodic solutions having the same Frequency Vector as the forcing term. Earlier results available in the literature show that, when the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions can be proved to exist and to be attractive provided some Diophantine condition is assumed on the Frequency Vector. In this paper we show that the results extend to the case of arbitrary Frequency Vectors.
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Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies
Journal of Differential Equations, 1Co-Authors: Guido Gentile, Faenia VaiaAbstract:Abstract We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same Frequency Vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non-resonance condition on the Frequency Vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors, we confine ourselves to two-dimensional Frequency Vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the Frequency Vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with “holes”. If we want the set to be connected we have to require some non-resonance condition on the Frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.
Petr Novotny - One of the best experts on this subject based on the ideXlab platform.
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long run average behaviour of probabilistic Vector addition systems
Logic in Computer Science, 2015Co-Authors: Tomáš Brázdil, Antonín Kučera, Stefan Kiefer, Petr NovotnyAbstract:We study the pattern Frequency Vector for runs in probabilistic Vector Addition Systems with States (pVASS). Intuitively, each configuration of a given pVASS is assigned one of finitely many patterns, and every run can thus be seen as an infinite sequence of these patterns. The pattern Frequency Vector assigns to each run the limit of pattern frequencies computed for longer and longer prefixes of the run. If the limit does not exist, then the Vector is undefined. We show that for one-counter pVASS, the pattern Frequency Vector is defined and takes one of finitely many values for almost all runs. Further, these values and their associated probabilities can be approximated up to an arbitrarily small relative error in polynomial time. For stable two-counter pVASS, we show the same result, but we do not provide any upper complexity bound. As a byproduct of our study, we discover counterexamples falsifying some classical results about stochastic Petri nets published in the 80s.
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Long-Run Average Behaviour of Probabilistic Vector Addition Systems
arXiv: Logic in Computer Science, 2015Co-Authors: Tomáš Brázdil, Antonín Kučera, Stefan Kiefer, Petr NovotnyAbstract:We study the pattern Frequency Vector for runs in probabilistic Vector Addition Systems with States (pVASS). Intuitively, each configuration of a given pVASS is assigned one of finitely many patterns, and every run can thus be seen as an infinite sequence of these patterns. The pattern Frequency Vector assigns to each run the limit of pattern frequencies computed for longer and longer prefixes of the run. If the limit does not exist, then the Vector is undefined. We show that for one-counter pVASS, the pattern Frequency Vector is defined and takes only finitely many values for almost all runs. Further, these values and their associated probabilities can be approximated up to an arbitrarily small relative error in polynomial time. For stable two-counter pVASS, we show the same result, but we do not provide any upper complexity bound. As a byproduct of our study, we discover counterexamples falsifying some classical results about stochastic Petri nets published in the~80s.
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LICS - Long-Run Average Behaviour of Probabilistic Vector Addition Systems
2015 30th Annual ACM IEEE Symposium on Logic in Computer Science, 2015Co-Authors: Tomáš Brázdil, Antonín Kučera, Stefan Kiefer, Petr NovotnyAbstract:We study the pattern Frequency Vector for runs in probabilistic Vector Addition Systems with States (pVASS). Intuitively, each configuration of a given pVASS is assigned one of finitely many patterns, and every run can thus be seen as an infinite sequence of these patterns. The pattern Frequency Vector assigns to each run the limit of pattern frequencies computed for longer and longer prefixes of the run. If the limit does not exist, then the Vector is undefined. We show that for one-counter pVASS, the pattern Frequency Vector is defined and takes one of finitely many values for almost all runs. Further, these values and their associated probabilities can be approximated up to an arbitrarily small relative error in polynomial time. For stable two-counter pVASS, we show the same result, but we do not provide any upper complexity bound. As a byproduct of our study, we discover counterexamples falsifying some classical results about stochastic Petri nets published in the 80s.
Livia Corsi - One of the best experts on this subject based on the ideXlab platform.
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Resonant tori of arbitrary codimension for quasi-periodically forced systems
Nonlinear Differential Equations and Applications NoDEA, 2016Co-Authors: Livia Corsi, Guido GentileAbstract:We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility property and we assume its Frequency Vector to be Bryuno. Then we prove that, without imposing any non-degeneracy condition on the forcing, there exists at least one quasi-periodic solution with the same Frequency Vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases.
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convergent series for quasi periodically forced strongly dissipative systems
Communications in Contemporary Mathematics, 2014Co-Authors: Livia Corsi, Roberto Feola, Guido GentileAbstract:We study the ordinary differential equation eẍ + ẋ + eg(x) = ef(ωt), with f and g analytic and f quasi-periodic in t with Frequency Vector ω ∈ ℝd. We show that if there exists c0 ∈ ℝ such that g(c0) equals the average of f and the first non-zero derivative of g at c0 is of odd order 𝔫, then, for e small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c0, with the same Frequency Vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for 𝔫 = 1. We also point out that, if 𝔫 = 1 and the first derivative of g at c0 is positive, then the quasi-periodic solution is locally unique and attractive.
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Domains of analyticity for response solutions in strongly dissipative forced systems
Journal of Mathematical Physics, 2013Co-Authors: Livia Corsi, Roberto Feola, Guido GentileAbstract:We study the ordinary differential equation ɛx+x+ɛg(x)=ɛf(ωt), where g and f are real-analytic functions, with f quasi-periodic in t with Frequency Vector ω. If c0∈R is such that g(c0) equals the average of f and g′(c0) ≠ 0, under very mild assumptions on ω there exists a quasi-periodic solution close to c0 with Frequency Vector ω. We show that such a solution depends analytically on ɛ in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.
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convergent series for quasi periodically forced strongly dissipative systems
arXiv: Dynamical Systems, 2012Co-Authors: Livia Corsi, Roberto Feola, Guido GentileAbstract:We study the ordinary differential equation ${\varepsilon}\ddot x+\dot x + {\varepsilon} g(x) = {\varepsilon} f(\omega t)$, with $f$ and $g$ analytic and $f$ quasi-periodic in $t$ with Frequency Vector $\omega\in R^{d}$. We show that if there exists $c_0\in R$ such that $g(c_0)$ equals the average of $f$ and the first non-zero derivative of $g$ at $c_0$ is of odd order $n$, then, for ${\varepsilon}$ small enough and under very mild Diophantine conditions on $\omega$, there exists a quasi-periodic solution close to $c_0$, with the same Frequency Vector as $f$. In particular if $f$ is a trigonometric polynomial the Diophantine condition on $\omega$ can be completely removed. This extends results previously available in the literature for $n=1$. We also point out that, if $n=1$ and the first derivative of $g$ at $c_0$ is positive, then the quasi-periodic solution is locally unique and attractive.
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response solutions for arbitrary quasi periodic perturbations with bryuno Frequency Vector
arXiv: Dynamical Systems, 2010Co-Authors: Livia Corsi, Guido GentileAbstract:We study the problem of existence of response solutions for a real-analytic one-dimensional system, consisting of a rotator subjected to a small quasi-periodic forcing. We prove that at least one response solution always exists, without any assumption on the forcing besides smallness and analyticity. This strengthens the results available in the literature, where generic non-degeneracy conditions are assumed. The proof is based on a diagrammatic formalism and relies on renormalisation group techniques, which exploit the formal analogy with problems of quantum field theory; a crucial role is played by remarkable identities between classes of diagrams.
Weng Cho Chew - One of the best experts on this subject based on the ideXlab platform.
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A Low Frequency Vector Fast Multipole Algorithm with Vector Addition Theorem
Communications in Computational Physics, 2010Co-Authors: Yang G. Liu, Weng Cho ChewAbstract:In the low-Frequency fast multipole algorithm (LF-FMA) [19, 20], scalar addition theorem has been used to factorize the scalar Green's function. Instead of this traditional factorization of the scalar Green's function by using scalar addition theorem, we adopt the Vector addition theorem for the factorization of the dyadic Green's function to realize memory savings. We are to validate this factorization and use it to develop a low-Frequency Vector fast multipole algorithm (LF-VFMA) for lowFrequency problems. In the calculation of non-near neighbor interactions, the storage of translators in the method is larger than that in the LF-FMA with scalar addition theorem. Fortunately it is independent of the number of unknowns. Meanwhile, the storage of radiation and receiving patterns is linearly dependent on the number of unknowns. Therefore it is worthwhile for large scale problems to reduce the storage of this part. In thismethod, the storage of radiation and receiving patterns can be reduced by 25 percent compared with the LF-FMA. © 2010 Global-Science Press.link_to_subscribed_fulltex
