Transport Properties of a Piecewise Linear Transformation and Deterministic Levy Flights

Miyaguchi, Tomoshige

doi:10.1143/ptp.115.31

Cited by 4 publications

(1 citation statement)

References 17 publications

(11 reference statements)

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“…Piecewise linear approximations are often used in the studies of nonlinear dynamical systems. In fact, even for systems in which rigorous approaches are difficult, more detailed analysis is possible for the piecewise linear versions [23][24][25][26]. Here, we derive the stable ranges of two-level and three-level solutions for the piecewise-linear model.…”

Section: Introductionmentioning

confidence: 99%

A piecewise linear model of self-organized hierarchy formation

Miyaguchi,

Miki,

Hamada

2020

Preprint

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The Bonabeau model of self-organized hierarchy formation is studied by using a piecewise linear approximation to the sigmoid function. Simulations of the piecewise-linear agent model show that there exist two-level and three-level hierarchical solutions, and that each agent exhibits a transition from non-ergodic to ergodic behaviors. Furthermore, by using a mean-field approximation to the agent model, it is analytically shown that there are asymmetric two-level solutions, even though the model equation is symmetric (asymmetry is introduced only through the initial conditions), and that linearly stable and unstable three-level solutions coexist. It is also shown that some of these solutions emerge through supercritical-pitchfork-like bifurcations in invariant subspaces. Existence and stability of the linear hierarchy solution in the mean-field model are also presented.

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Section: Introductionmentioning

confidence: 99%

A piecewise linear model of self-organized hierarchy formation

Miyaguchi,

Miki,

Hamada

2020

Preprint

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Piecewise linear model of self-organized hierarchy formation

2020

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Topology of magnetic field lines: Chaos and bifurcations emerging from two-action systems

et al. 2011

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Nonlinear dynamics of magnetic field lines generated by simple electric current elements are investigated. In general, the magnetic field lines show behavior similar to that of the Hamiltonian systems; in fact, they can be generally transformed into Hamiltonian systems with 1.5 degrees of freedom, obey the Kolmogorov-Arnold-Moser (KAM) theorem, and generate chaotic trajectories. In the case where unperturbed systems are described by two action (slow) and one angle (fast) variables, however, it is found that the periodic orbits of the unperturbed systems vanish for arbitrarily small symmetry-breaking perturbations (a breakdown of the KAM theorem) and drifting or periodic trajectories appear. The mechanism of this phenomenon is investigated analytically by weak nonlinear stability analysis. It is also shown numerically that scattering processes of the perturbed system exhibit typical features of chaotic dynamical systems.

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Transport Properties of a Piecewise Linear Transformation and Deterministic Levy Flights

Cited by 4 publications

References 17 publications

A piecewise linear model of self-organized hierarchy formation

A piecewise linear model of self-organized hierarchy formation

Piecewise linear model of self-organized hierarchy formation

Topology of magnetic field lines: Chaos and bifurcations emerging from two-action systems

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