# Nonlinear Sciences

## New submissions

[ total of 7 entries: 1-7 ]
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### New submissions for Fri, 16 Feb 18

[1]
Title: Extensive numerical study and circuitry implementation of the Watt governor model
Journal-ref: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, v. 27, p. 1750175, 2017
Subjects: Chaotic Dynamics (nlin.CD)

In this work we carry out extensive numerical study of a Watt-centrifugal-governor system model, and we also implement an electronic circuit by analog computation to experimentally solve the model. Our numerical results show the existence of self-organized stable periodic structures (SPSs) on parameter-space of the largest Lyapunov exponent and isospikes of time series of the Watt governor system model. A peculiar hierarchical organization and period-adding bifurcation cascade of the SPSs are observed, and this self-organized cascade accumulates on a periodic boundary. It is also shown that the periods of these structures organize themselves obeying the solutions of Diophantine equations. In addition, an experimental setup is implemented by a circuitry analogy of mechanical systems using analog computing technique to characterize the robustness of our numerical results. After applying an active control of chaos in the experiment, the effect of intrinsic experimental noise was minimized such that, the experimental results are in astonishing well agreement with our numerical findings. We can also mention as another remarkable result, the application of analog computing technique to perform an experimental circuitry analysis in real mechanical problems.

[2]
Title: Chaos in Kuramoto Oscillator Networks
Subjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS); Adaptation and Self-Organizing Systems (nlin.AO)

Kuramoto oscillators are widely used to explain collective phenomena in networks of coupled oscillatory units. We show that simple networks of two populations with a generic coupling scheme can exhibit chaotic dynamics as conjectured by Ott and Antonsen [Chaos, 18, 037113 (2008)]. These chaotic mean field dynamics arise universally across network size, from the continuum limit of infinitely many oscillators down to very small networks with just two oscillators per population. Hence, complicated dynamics are expected even in the simplest description of oscillator networks.

[3]
Title: Dynamics of high-order solitons in the nonlocal nonlinear Schrödinger equations
Authors: Bo Yang, Yong Chen
Subjects: Pattern Formation and Solitons (nlin.PS)

A study of high-order solitons in three nonlocal nonlinear Schr\"{o}dinger equations is presented, which includes the \PT-symmetric, reverse-time, and reverse-space-time nonlocal nonlinear Schr\"{o}dinger equations. General high-order solitons in three different equations are derived from the same Riemann-Hilbert solutions of the AKNS hierarchy, except for the difference in the corresponding symmetry relations on the "perturbed" scattering data. Dynamics of general high-order solitons in these equations is further analyzed. It is shown that the high-order fundamental-soliton is always moving on several different trajectories in nearly equal velocities, and they can be nonsingular or repeatedly collapsing, depending on the choices of the parameters. It is also shown that high-order multi-solitons could have more complicated wave structures and behave very differently from high-order fundamental solitons. More interesting is the high-order hybrid-pattern solitons, which are derived from combination of different size of block matrix in the Riemann-Hilbert solutions and thus they can describe a nonlinear interaction between several types of solitons.

### Cross-lists for Fri, 16 Feb 18

[4]  arXiv:1802.05389 (cross-list from physics.soc-ph) [pdf, other]
Title: Dynamical Galam model
Comments: LaTeX, 14 pages, 7 figues
Subjects: Physics and Society (physics.soc-ph); Adaptation and Self-Organizing Systems (nlin.AO)

We introduce a model of temporal evolution of political opinions which amounts to a dynamical extension of Galam model in which the number of inflexibles are treated as dynamical variables. We find that the critical value of inflexibles in the original Galam model now turns into a fixed point of the system whose stability controls the phase trajectory of the political opinions. The appearance of two phases, in which majority-preserving and regime-changing limit cycles are respectively dominant, is found, and also the transition between them is observed.

[5]  arXiv:1802.05393 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
Title: Unusual Anomalous Heat Spread Contributed by Optical Phonon-Phonon Interactions
Subjects: Statistical Mechanics (cond-mat.stat-mech); Chaotic Dynamics (nlin.CD)

We employ the equilibrium correlation functions to study heat spread in an anharmonic chain subject to a \emph{harmonic} pinned potential. Such a system is nonacoustic with purely deterministic dynamics, while previously, due to the momentum-nonconserving feature, it was naturally thought to follow the normal diffusive transport. Here, instead, we find that, the heat spread is superdiffusive, yet implying the anomalous transport. In particular, a non-Gaussian truncated L\'{e}vy walk heat spreading function with anomalous scaling is, for the first time, observed, and the underlying mechanism is related to a novel effect of optical phonon-phonon interactions. As most of the current theories for deterministic dynamics are mainly based on the long-wavelength acoustic phonons, and for the pinned systems normal transport was always predicted, our results may open new avenues for further exploring thermal transport.

[6]  arXiv:1802.05460 (cross-list from math.CA) [pdf, other]
Title: Shape invariance and equivalence relations for pseudowronskians of Laguerre and Jacobi polynomials
Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

In a previous paper we derived equivalence relations for pseudo-Wronskian determinants of Hermite polynomials. In this paper we obtain the analogous result for Laguerre and Jacobi polynomials. The equivalence formulas are richer in this case since rational Darboux transformations can be defined for four families of seed functions, as opposed to only two families in the Hermite case. The pseudo-Wronskian determinants of Laguerre and Jacobi type will thus depend on two Maya diagrams, while Hermite pseudo-Wronskians depend on just one Maya diagram. We show that these equivalence relations can be interpreted as the general transcription of shape invariance and specific discrete symmetries acting on the parameters of the isotonic oscillator and Darboux-Poschl-Teller potential.

[7]  arXiv:1802.05547 (cross-list from math.AP) [pdf, ps, other]
Title: Dynamics of small solutions in KdV type equations: decay inside the linearly dominated region
Authors: Claudio Muñoz
In this paper we prove that all small, uniformly in time $L^1\cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $x\ll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime. For the proof, we make use of well-chosen weighted virial identities. The main new idea employed here with respect to previous results is the fact that the $L^1$ integral is subcritical with respect to the KdV scaling.