scholarly journals Safe basins for a nonlinear oscillator with ramped forcing

2016 ◽  
Vol 472 (2194) ◽  
pp. 20160190 ◽  
Author(s):  
James A. Wright
Keyword(s):  
Nonlinear Oscillator ◽  
Basins Of Attraction ◽  
Previous Literature ◽  
External Forcing ◽  
Safe Basin

A nonlinear oscillator is studied in the presence of external forcing for which the amplitude initially depends on time. The focus is on the sizes of the basins of attraction which do not lead to unbounded motions, collectively termed the ‘safe basin’. Direct comparisons are drawn between the regime of constant forcing amplitude and that where the forcing amplitude initially depends on time. In the process, questions from previous literature are answered and previously unexplained phenomena are understood. Furthermore, we witness a new phenomenon, not previously observed for the system studied.

Sequences of Global Bifurcations and the Related Outcomes after Crisis of the Resonant Attractor in a Nonlinear Oscillator

1997 ◽  
Vol 07 (11) ◽  
pp. 2437-2457 ◽  
Author(s):  
W. Szemplińska-Stupnicka ◽  
E. Tyrkiel
Keyword(s):  
Nonlinear Oscillator ◽  
Duffing Oscillator ◽  
Nonlinear Resonance ◽  
Basins Of Attraction ◽  
Global Bifurcations ◽  
System Behavior ◽  
Coexisting Attractors ◽  
System Parameters

The problem of the system behavior after annihilation of the resonant attractor in the region of the nonlinear resonance hysteresis is considered. The sequences of global bifurcations, in connection with the associated metamorphoses of basins of attraction of coexisting attractors, are examined. The study allows one to reveal the mechanism that governs the phenomenon of the post crisis ensuing transient trajectory to settle onto one or another remote attractor. The problem is studied in detail for the twin-well potential Duffing oscillator. The boundary which splits the considered region of system parameters into two subdomains, where the outcome is unique or the two outcomes are possible, is defined.

scholarly journals Dynamical Tangles in Third-Order Oscillator with Single Jump Function

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Jiří Petržela ◽  
Tomas Gotthans ◽  
Milan Guzan
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Nonlinear Oscillator ◽  
Invariant Manifolds ◽  
Order Differential Equation ◽  
Dynamical Behavior ◽  
Basins Of Attraction ◽  
Third Order ◽  
Newtonian Dynamics ◽  
Jump Function

This contribution brings a deep and detailed study of the dynamical behavior associated with nonlinear oscillator described by a single third-order differential equation with scalar jump nonlinearity. The relative primitive geometry of the vector field allows making an exhaustive numerical analysis of its possible solutions, visualizations of the invariant manifolds, and basins of attraction as well as proving the existence of chaotic motion by using the concept of both Shilnikov theorems. The aim of this paper is also to complete, carry out and link the previous works on simple Newtonian dynamics, and answer the question how individual types of the phenomenon evolve with time via understandable notes.

Analysis on an Oscillator With Delayed Feedback Near Double Hopf Bifurcations

2001 ◽  
Author(s):  
P. Yu ◽  
Y. Yuan
Keyword(s):  
Time Delay ◽  
Nonlinear Oscillator ◽  
Analytical Approach ◽  
Complex Dynamics ◽  
Integration Scheme ◽  
Hopf Bifurcations ◽  
External Forcing ◽  
Chaotic Motions ◽  
System Parameters ◽  
Numerical Integration Scheme

Abstract This paper considers the effect of time delayed feedbacks in a nonlinear oscillator with external forcing. The particular attention is focused on the case where the corresponding linear system has two pairs of purely imaginary eigenvalues at a critical point, leading to double Hopf bifurcations. An analytical approach is used to find the explicit expressions for the critical values of the system parameters at which non-resonant or resonant Hopf bifurcations may occur. A fourth-order Ronge-Kutta numerical integration scheme is applied to obtain the dynamical solutions in the vicinity of the critical points. Both the cases with and without the external forcing are considered. It has been found the system exhibits very rich complex dynamics including periodic, quasi-periodic and chaotic motions. Moreover, a sensitivity analysis is carried out to show that chaotic motions are very sensitive to the time delay. This suggests that the time delay can be used: (1) to control bifurcations and chaos; and (2) to generate bifurcations and chaos.

Analysis of Bifurcations and Chaos in Three Coupled Physical Pendulums With Impacts

2001 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Grzegorz Kudra ◽  
C.-H. Lamarque
Keyword(s):  
Lyapunov Exponents ◽  
Discontinuity Point ◽  
Basins Of Attraction ◽  
External Forcing ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Transition Condition ◽  
Triple Pendulum ◽  
Linearized System

Abstract In this work the triple pendulum with damping, external forcing and with impacts is investigated. The extension of a coefficient restitution rule and a special transition condition rule for perturbation (linearized system in the Lyapunov exponents algorithm) in each discontinuity point are applied. Periodic, quasi-periodic, chaotic and hyperchaotic motions are observed using Poincaré maps and bifurcational diagrams, which are verified by the Lyapunov exponents. In additon basins of attraction of some coexisting regular and irregular attractors are illustrated and discussed.

PHASE MODEL REDUCTION AND SYNCHRONIZATION OF PERIODICALLY FORCED NONLINEAR OSCILLATORS

2010 ◽  
Vol 19 (04) ◽  
pp. 749-762 ◽  
Author(s):  
MICHELE BONNIN ◽  
FERNANDO CORINTO ◽  
MARCO GILLI
Keyword(s):  
Physical System ◽  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Phase Model ◽  
External Forcing ◽  
State Equations ◽  
Phase Models ◽  
Ideal Framework ◽  
Averaging Techniques ◽  
The Ideal

Phase models represent the ideal framework to investigate the synchronization of a nonlinear oscillator with an external forcing. While many researches focused the attention to their analysis, little work has been done about the reduction of a physical system to the corresponding phase model. In this paper we show how, resorting to averaging techniques, it is possible to obtain the phase model corresponding to a given set of state equations. As examples, we derive the phase equations and investigate the synchronization properties of two popular nonlinear oscillators.

Interactions Between Two Coupled Nonlinear Forced Systems: Fast/Slow Dynamics

2016 ◽  
Vol 26 (09) ◽  
pp. 1650155 ◽  
Author(s):  
S. Charlemagne ◽  
A. Ture Savadkoohi ◽  
C.-H. Lamarque
Keyword(s):  
Nonlinear Oscillator ◽  
Complex Geometry ◽  
Equilibrium Points ◽  
Linear Structure ◽  
Singular Points ◽  
Multiple Time ◽  
External Forcing ◽  
Multiple Time Scale ◽  
Direct Integration ◽  
Structural System

Dynamics of a system formed by a linear structure coupled to a light nonlinear oscillator, both subjected to external excitations, is studied. Effects of the external forcing of the nonlinear oscillator are especially investigated. Complex geometry of the slow invariant manifold and equilibrium and singular points of the system are detected thanks to a multiple time scale strategy around 1:1:1 resonance. Equilibrium points lead to periodic regimes while singular points are hints of strongly modulated response of the system characterized by repeated bifurcations around its stable zones. A method for detection of changes in mechanical properties of main structural system is explained. Numerical simulations obtained by direct integration of the system are used to validate analytical predictions.

Non-Lipschitzian Control Algorithm for Nanoscale

2003 ◽  
Vol 790 ◽  
Author(s):  
Friction V. Protopopescu ◽  
J. Barhen ◽  
Y. Braiman
Keyword(s):  
Nonlinear Oscillator ◽  
Control Algorithm ◽  
System Size ◽  
Basins Of Attraction ◽  
Controlled System ◽  
Control Approach ◽  
Strong Robustness ◽  
Nonlinear Oscillator Array ◽  
Short Times ◽  
Kontorova Model

We present a robust feedback control algorithm and apply it to the nonlinear oscillator array (Frenkel-Kontorova) model of nanoscale friction. The new control approach is based on the concepts of non-Lipschitzian dynamics and global targeting. We show that average quantities of the controlled system can be driven - exactly or approximately - towards desired targets which become additional, linearly stable attractors for the system's dynamics. Extensive numerical simulations show that the basins of attraction of these targets are reached in very short times and the control exhibits very strong robustness. We investigate the efficiency of the control in terms of various parameters (e.g., system size, non-Lipschitzian exponent).

The Elachistidae of southern Siberia and Central Asia, with descriptions of five new species (Lepidoptera)¹

1992 ◽  
Vol 3 (4) ◽  
pp. 177-194 ◽  
Author(s):  
Lauri Kaila
Keyword(s):  
New Species ◽  
Central Asia ◽  
Baikal Region ◽  
Tianshan Mountains ◽  
Previous Literature ◽  
Altai Mountains ◽  
Southern Siberia

The Elachistidae material collected during the joint Soviet-Finnish entomological expeditions to the Altai mountains, Baikal region and Tianshan mountains of the previous USSR is listed. Previous literature dealing with the Elachistidae in Central Asia is reviewed. A total of 40 species are dealt with, including descriptions of five new species: Stephensia jalmarella sp. n. (Altai), Elachista baikalica sp. n. (Baikal), E. talgarella sp. n. (southern Kazakhstan), E. esmeralda sp. n. (southern Kazakhstan) and E. filicornella sp. n. (southern Kazakhstan). The previously unknown females of E. bimaculata Parenti, 1981 and Biselachista zonulae Sruoga, 1992 are described.

Intuitive Biology Thinking: Comparing Chinese and US Middle Schoolers

2020 ◽  
Author(s):  
Yian Xu ◽  
John Coley
Keyword(s):  
Previous Literature ◽  
Western Population ◽  
Middle Schoolers ◽  
The Us ◽  
Ways Of Thinking ◽  
Cognitive Frameworks ◽  
8Th Graders ◽  
Universal Nature

Previous literature demonstrated that people spontaneously engage in systematic ways of thinking about biology. However, with most studies focusing on the western population, little is known about the universal nature of these cognitive frameworks. The current study used a construal-based survey to systematically test intuitive biology thinking in China. Overall, Chinese 8th graders demonstrated stronger essentialist thinking, weaker anthropocentric thinking, and similar level of teleological thinking compared to the US counterparts.

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