scholarly journals HAMILTONIAN-BASED FREQUENCY-AMPLITUDE FORMULATION FOR NONLINEAR OSCILLATORS

2021 ◽  
Vol 19 (2) ◽  
pp. 199 ◽  
Author(s):  
Ji-Huan He ◽  
Wei-Fan Hou ◽  
Na Qie ◽  
Khaled A. Gepreel ◽  
Ali Heidari Shirazi ◽  
...  
Keyword(s):  
Nonlinear Dynamics ◽  
Energy Conservation ◽  
Variational Formulation ◽  
Nonlinear Oscillator ◽  
Solution Process ◽  
Nonlinear Oscillators ◽  
Nonlinear Interactions ◽  
Conservative Oscillator ◽  
Vibration Process ◽  
Complex Mechanical Systems

Complex mechanical systems usually include nonlinear interactions between their components which can be modeled by nonlinear equations describing the sophisticated motion of the system. In order to interpret the nonlinear dynamics of these systems, it is necessary to compute more precisely their nonlinear frequencies. The nonlinear vibration process of a conservative oscillator always follows the law of energy conservation. A variational formulation is constructed and its Hamiltonian invariant is obtained. This paper suggests a Hamiltonian-based formulation to quickly determine the frequency property of the nonlinear oscillator. An example is given to explicate the solution process.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

PERIODIC INHIBITION OF LIVING PACEMAKER NEURONS (I): LOCKED, INTERMITTENT, MESSY, AND HOPPING BEHAVIORS

1991 ◽  
Vol 01 (03) ◽  
pp. 549-581 ◽  
Author(s):  
J. P. SEGUNDO ◽  
E. ALTSHULER ◽  
M. STIBER ◽  
A. GARFINKEL
Keyword(s):  
Nonlinear Dynamics ◽  
Point Process ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Strange Attractors ◽  
Synaptic Inhibition ◽  
Pacemaker Neurons ◽  
Pacemaker Neuron ◽  
Exhaustive List

This communication is concerned with an embodiment of periodic nonlinear oscillator driving, the synaptic inhibition of one spike-producing pacemaker neuron by another. Data came from a prototypical living synapse. Analyses centered on a prolonged condition between the transients following the onset and cessation of inhibition. Evaluations were guided by point process mathematics and nonlinear dynamics. A rich and exhaustive list of discharge forms, described precisely and canonically, was observed across different inhibitory rates. Previously unrecognized at synapses, most forms were identified with several well known types from nonlinear dynamics. Ordered by decreasing regularities, they were locked, intermittent (including walk-throughs), messy (including erratic and stammerings) and hopping. Each is discussed within physiological and formal contexts. It is conjectured that (i) locked, intermittent and messy forms reflect limit cycles on 2-tori, quasiperiodic orbits and strange attractors, (ii) noise in neurons hovering around threshold contributes to certain intermittent and stammering behaviors, and (iii) hopping either reflects an attractor with several portions or is nonstationary and noise-induced.

NONLINEAR DYNAMICS OF DIGITAL PHASE-LOCKED LOOPS WITH DELAY

1994 ◽  
Vol 04 (03) ◽  
pp. 715-726 ◽  
Author(s):  
MARIA DE SOUSA VIEIRA ◽  
ALLAN J. LICHTENBERG ◽  
MICHAEL A. LIEBERMAN
Keyword(s):  
Nonlinear Dynamics ◽  
Chaotic Behavior ◽  
Nonlinear Oscillators ◽  
Coupled Systems ◽  
Phase Locked Loops ◽  
Bifurcation Diagrams ◽  
Digital Phase ◽  
Coupled Nonlinear Oscillators ◽  
Digital Phase Locked Loops ◽  
The Stability

We investigate numerically and analytically the nonlinear dynamics of a system consisting of two self-synchronizing pulse-coupled nonlinear oscillators with delay. The particular system considered consists of connected digital phase-locked loops. We find mapping equations that govern the system and determine the synchronization properties. We study the bifurcation diagrams, which show regions of periodic, quasiperiodic and chaotic behavior, with unusual bifurcation diagrams, depending on the delay. We show that depending on the parameter that is varied, the delay will have a synchronizing or desynchronizing effect on the locked state. The stability of the system is studied by determining the Liapunov exponents, indicating marked differences compared to coupled systems without delay.

scholarly journals Approximate Potentials with Applications to Strongly Nonlinear Oscillators with Slowly Varying Parameters

2003 ◽  
Vol 10 (5-6) ◽  
pp. 379-386 ◽  
Author(s):  
Jianping Cai ◽  
Y.P. Li
Keyword(s):  
Present Method ◽  
Nonlinear Oscillator ◽  
Elliptic Functions ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Elapsed Time ◽  
Strongly Nonlinear Oscillators ◽  
Strongly Nonlinear Oscillator

A method of approximate potential is presented for the study of certain kinds of strongly nonlinear oscillators. This method is to express the potential for an oscillatory system by a polynomial of degree four such that the leading approximation may be derived in terms of elliptic functions. The advantage of present method is that it is valid for relatively large oscillations. As an application, the elapsed time of periodic motion of a strongly nonlinear oscillator with slowly varying parameters is studied in detail. Comparisons are made with other methods to assess the accuracy of the present method.

scholarly journals Quantum steering in an asymmetric chain of nonlinear oscillators

2017 ◽  
Vol 9 (3) ◽  
pp. 97 ◽  
Author(s):  
Joanna K. Kalaga ◽  
Wiesław Leoński ◽  
Radosław Szczęśniak
Keyword(s):  
Solid State ◽  
Superconducting State ◽  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Quantum Spin ◽  
Uncertainty Relations ◽  
Quantum Spin Systems ◽  
Optomechanical System ◽  
Einstein Podolsky Rosen ◽  
Quantum Steering

We discuss here a possibility of generation of steerable states in asymmetric chains comprising three Kerr-like nonlinear oscillators. We show that steering between modes can be generated in the system and it strongly depends on the asymmetry of internal couplings in our model. We can lead to the appearance of new steering effects, which were not present in symmetric models already studied in the literature. Full Text: PDF ReferencesE. Schrödinger, "Discussion of Probability Relations between Separated Systems", Math. Proc. Camb. Phil. Soc. 31, 555 (1935). CrossRef M.D. Reid, "Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification", Phys. Rev. A 40, 913 (1989). CrossRef E.G. Cavalcanti, M.D. Reid, "Uncertainty relations for the realization of macroscopic quantum superpositions and EPR paradoxes", Journal of Modern Optics 54, 2373 (2007). CrossRef S.P. Walborn, A. Salles, R.M. Gomes, F. Toscano, P.H. Souto Ribeiro, "Revealing Hidden Einstein-Podolsky-Rosen Nonlocality", Phys. Rev. Lett. 106, 130402 (2011). CrossRef H.M. Wiseman, S.J. Jones, A.C. Doherty, "Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox", Phys. Rev. Lett. 98, 140402 (2007). CrossRef S.J. Jones, H.M. Wiseman, A.C. Doherty, "Entanglement, Einstein-Podolsky-Rosen correlations, Bell nonlocality, and steering", Phys. Rev. A 76, 052116 (2007). CrossRef J.K. Kalaga, W. Leoński, "Quantum steering borders in three-qubit systems", Quantum Inf Process 16, 175 (2017). CrossRef Q. He, Z. Ficek, "Einstein-Podolsky-Rosen paradox and quantum steering in a three-mode optomechanical system", Phys. Rev. A 89, 022332 (2014). CrossRef S. Kiesewetter, Q.Y. He, P.D. Drummond, M.D. Reid, "Scalable quantum simulation of pulsed entanglement and Einstein-Podolsky-Rosen steering in optomechanics", Phys. Rev. A 90, 043805 (2014). CrossRef K. Bartkiewicz, A. Cernoch, K. Lemr, A. Miranowicz, F. Nori, "Experimental temporal quantum steering", Scientific Reports 6, 38076 (2016). CrossRef A. Barasiński, B. Brzostowski, R. Matysiak, P. Sobczak, D. Woźniak, In: R. Wyrzykowski, J. Dongarra, K. Karczewski, J. Wasniewski editor, Parallel Processing and Applied Mathematics (PPAM 2013), Lecture Notes in Computer Science, vol 8385. Springer, Berlin, Heidelberg (2014). CrossRef A. Drzewiński, J. Sznajd, "On the real-space renormalization-group study of some 2D quantum spin systems", Physica A 170, 415 (1991). CrossRef G.J. Milburn, C.A. Holmes, "Quantum coherence and classical chaos in a pulsed parametric oscillator with a Kerr nonlinearity", Phys. Rev. A 44, 4704 (1991). CrossRef W. Leoński, "Quantum and classical dynamics for a pulsed nonlinear oscillator", Physica A 233, 365 (1996). CrossRef A. Kowalewska-Kudłaszyk, J.K. Kalaga, W. Leoński, "Long-time fidelity and chaos for a kicked nonlinear oscillator system", Physics Letters A 373, 1334 (2009). CrossRef J.K. Kalaga, W. Leoński, "Two proposals of quantum chaos indicators related to the mean number of photons: pulsed Kerr-like oscillator case", Proc. SPIE 10142, 101421L (2016). CrossRef A. Barasiński, W. Leoński, T. Sowiński, "Ground-state entanglement of spin-1 bosons undergoing superexchange interactions in optical superlattices", J. Opt. Soc. Am. B 31, 1845 (2014). CrossRef A. Barasiński, W. Leoński, "Symmetry restoring and ancilla-driven entanglement for ultra-cold spin-1 atoms in a three-site ring", Quantum Inf Process 16, 6 (2017). CrossRef D. Woźniak, A. Drzewiński, G. Kamieniarz, "Relaxation Dynamics in the Spin-1 Heisenberg Antiferromagnetic Chain after a Quantum Quench of the Uniaxial Anisotropy", Acta Physica Polonica A 130, 1395 (2016). CrossRef R. Szczęśniak, D. Szczęśniak, E.A. Drzazga, "Superconducting state in the atomic metallic hydrogen just above the pressure of the molecular dissociation", Solid State Communications 152, 2023 (2012). CrossRef A. P. Durajski, R. Szczęśniak, M.W. Jarosik, "Properties of the superconducting state in compressed sulphur", Phase Transitions 85, 727 (2012). CrossRef R. Szczęśniak, A. P. Durajski, "The thermodynamic properties of the high-pressure superconducting state in the hydrogen-rich compounds", Solid State Sciences 25, 45 (2013). CrossRef X. Wang, A. Miranowicz, H.R. Li, F. Nori, "Multiple-output microwave single-photon source using superconducting circuits with longitudinal and transverse couplings", Phys. Rev. A 94, 053858, (2016). CrossRef Y.X. Liu, X.W. Xu, A. Miranowicz, F. Nori, "From blockade to transparency: Controllable photon transmission through a circuit-QED system", Phys. Rev. A 89, 043818 (2014). CrossRef M.K. Olsen, "Spreading of entanglement and steering along small Bose-Hubbard chains", Phys. Rev. A 92, 033627 (2015). CrossRef E.G. Cavalcanti, Q.Y. He, M.D. Reid, H.M. Wiseman, "Unified criteria for multipartite quantum nonlocality", Phys. Rev. A 84, 032115 (2011). CrossRef

Nonlinear Oscillator-Based Gait Generation for a Novel Aero-Terrestrial Bioinspired Robotic System

2020 ◽  
Vol 12 (6) ◽  
Author(s):  
C. Alberto Sánchez-Delgado ◽  
Juan Carlos Ávila Vilchis ◽  
Adriana H. Vilchis-González ◽  
Belem Saldivar
Keyword(s):  
Gait Speed ◽  
Nonlinear Oscillator ◽  
Degrees Of Freedom ◽  
Nonlinear Oscillators ◽  
Robotic System ◽  
Simple Design ◽  
Flight Phase ◽  
Terrestrial Locomotion ◽  
Experimental Platform ◽  
Smooth Transitions

Abstract This paper focuses on the design of a novel aero-terrestrial robotic system based on the morphology of the Hymenoptera order insects and, particularly, on a strategy based on nonlinear oscillators for the coordination of its 12 terrestrial degrees-of-freedom (DoF). The ability of this new aero-terrestrial robot to, successfully, perform the walking process is validated through numerical simulations and tests performed on an experimental platform in which the gait speed was varied from 0.04 to 0.2 m/s. Some of the most important qualities of this robotic system are a relatively simple design with only 2 DoF per leg and a versatile terrestrial locomotion with the ability to vary its speed and direction in real-time with smooth transitions. Furthermore, unlike existent similar systems, the robot is designed to initiate a flight phase in any position without adopting particular postures avoiding undesirable interferences with the walking configuration.

Chaos Theory and Complexity Theory

2013 ◽  
Author(s):  
Keith Warren
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Systems Theory ◽  
Complexity Theory ◽  
Chaos Theory ◽  
Mathematical Framework ◽  
Nonlinear Interactions ◽  
Simple Systems ◽  
Over Time

Chaos theory and complexity theory, collectively known as nonlinear dynamics or dynamical systems theory, provide a mathematical framework for thinking about change over time. Chaos theory seeks an understanding of simple systems that may change in a sudden, unexpected, or irregular way. Complexity theory focuses on complex systems involving numerous interacting parts, which often give rise to unexpected order. The framework that encompasses both theories is one of nonlinear interactions between variables that give rise to outcomes that are not easily predictable. This entry provides a nonmathematical introduction, discussion of current research, and references for further reading.

Accurate numerical solutions of conservative nonlinear oscillators

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Khan Nasir Uddin ◽  
Khan Nadeem Alam
Keyword(s):  
Differential Equation ◽  
Plasma Physics ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Arbitrary Power

AbstractThe objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.

Nonlinear Dynamics in the Ultradian Rhythm of Desmodium motorium

1995 ◽  
Vol 50 (12) ◽  
pp. 1113-1116 ◽  
Author(s):  
Jyh-Phen Chen ◽  
Wolfgang Engelmann ◽  
Gerold Baier
Keyword(s):  
Nonlinear Dynamics ◽  
Time Series ◽  
Nonlinear Oscillators ◽  
Ultradian Rhythm ◽  
Ultradian Rhythms ◽  
Lateral Leaflet ◽  
Homoclinic Chaos ◽  
Coupled Nonlinear Oscillators

Abstract The dynamics of the lateral leaflet movement of Desmodium motorium is studied. Simple periodic, quasiperiodic and aperiodic time series are observed. The long-scale dynamics may either be uniform or composed of several prototypic oscillations (one of them reminiscent of homoclinic chaos). Diffusively coupled nonlinear oscillators may account for the variety of ultradian rhythms.

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