Nonlinear dynamics in meso and nano scales: fundamental aspects and applications

Luz, M. G. E. da; Anteneodo, Celia

doi:10.1098/rsta.2010.0301

Cited by 5 publications

(3 citation statements)

References 86 publications

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“…For instance, an operational approach is to focus on the expected purely technical (even formal) properties of complexity [55,56,103]. Then, one might search for minimal instances-either based on idealized physical models [104,105] or originated from plain mathematical constructions [106,107]-which could constitute prototypes examples for more realistic natural processes, and where characteristics commonly ascribed to actual complex systems (say, affinity, self-organization, feedback loops, hierarchical organization, etc [40,102,108]) may come about.…”

Section: Discussionmentioning

confidence: 99%

Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

Polli,

Raposo,

Viswanathan

et al. 2024

J. Phys. Complex.

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The numerical hailstone sequences, or orbits, generated by the Collatzmap have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n, the Collatz map f reads f(n) = n/2 ( f(n) = 3n + 1) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended ﬂuctuation, auto-correlation and entropy measure. Besides conﬁrming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our ﬁndings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical systemwhich could also be regarded as an arithmetic statistical physics system, explaining its dynamical richness.

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Section: Discussionmentioning

confidence: 99%

Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

Polli,

Raposo,

Viswanathan

et al. 2024

J. Phys. Complex.

View full text Add to dashboard Cite

show abstract

“…Looking at general processes, with inspiration coming from this principle, may also provide a new insight to the relation of classical and quantum chaos 28 , suggesting that classical deterministic chaos may be just an approximation with a characteristic time scale. Classical deterministic chaotic systems explode quickly, and observing the deterministic evolution of the system, even at a macroscopic level, enables the determination of the initial conditions increasingly precisely 29 .…”

Section: Discussionmentioning

confidence: 99%

Exponential Sensitivity and its Cost in Quantum Physics

Gilyén

Kiss

Jex

2016

Sci Rep

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State selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.

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“…[34] suggest that it is possible to fabricate cantilever tips coated by a magnetic film with a curvature radius in the range of 50 − 70 nm with maximum magnetic fields in the range of 40 − 80 mT. Due to a growing interest in complex behavior in nanodevices [35], the nonlinear and nonplanar phenomena exhibited by the system studied here have potential applications in signal processing, chaotic encryptation and random number generation [24,25].…”

Section: Discussionmentioning

confidence: 99%

Dynamics of a suspended nanowire driven by an ac Josephson current in an inhomogeneous magnetic field

Peña-Aza

2012

Phys. Rev. E

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We consider a voltage-biased nanoelectromechanical Josephson junction, where a suspended nanowire forms a superconducting weak-link, in an inhomogeneous magnetic field. We show that a nonlinear coupling between the Josephson current and the magnetic field generates a Laplace force that induces a whirling motion of the nanowire. By performing an analytical and a numerical analysis, we demonstrate that at resonance, the amplitude-phase dynamics of the whirling movement present different regimes depending on the degree of inhomogeneity of the magnetic field: time independent, periodic and chaotic. Transitions between these regimes are also discussed.

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Nonlinear dynamics in meso and nano scales: fundamental aspects and applications

Cited by 5 publications

References 86 publications

Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

Stochastic-like characteristics of arithmetic dynamical systems: the Collatz hailstone sequences

Exponential Sensitivity and its Cost in Quantum Physics

Dynamics of a suspended nanowire driven by an ac Josephson current in an inhomogeneous magnetic field

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