scholarly journals Bifurcation analysis of a vibro-impact experimental rig with two-sided constraint

Meccanica ◽  
2020 ◽  
Vol 55 (12) ◽  
pp. 2505-2521 ◽  
Author(s):  
Yang Liu ◽  
Joseph Páez Chávez ◽  
Bingyong Guo ◽  
Rauf Birler
Keyword(s):  
Bifurcation Analysis ◽  
Period Doubling ◽  
Path Following ◽  
External Excitation ◽  
Directional Control ◽  
Period 2 ◽  
Rectangular Waveform ◽  
Nonsmooth Dynamical Systems ◽  
Path Following Methods ◽  
Period Doubling Bifurcations

AbstractIn this paper we carry out an in-depth experimental and numerical investigation of a vibro-impact rig with a two-sided constraint and an external excitation given by a rectangular waveform. The rig, presenting forward and backward drifts, consists of an inner vibrating shaft intermittently impacting with its holding frame. Our interests focus on the multistability and the bifurcation structure observed in the system under two different contacting surfaces. For this purpose, we propose a mathematical model describing the rig dynamics and perform a detailed bifurcation analysis via path-following methods for nonsmooth dynamical systems, using the continuation platform COCO. Our study shows that multistability is produced by the interplay between two fold bifurcations, which give rise to hysteresis in the system. The investigation also reveals the presence of period-doubling bifurcations of limit cycles, which in turn are responsible for the creation of period-2 solutions for which the rig reverses its direction of progression. Furthermore, our study considers a two-parameter bifurcation analysis focusing on directional control, using the period of external excitation and the duty cycle of the rectangular waveform as the main control parameters.

SYNCHRONY-BREAKING PERIOD-DOUBLING BIFURCATIONS FOR THREE-CELL HOMOGENEOUS COUPLED MAPS

2009 ◽  
Vol 19 (04) ◽  
pp. 1157-1167
Author(s):  
ADELA COMANICI
Keyword(s):  
Fixed Points ◽  
Network Architecture ◽  
Vector Fields ◽  
Period Doubling ◽  
Feed Forward ◽  
Codimension One ◽  
Period 2 ◽  
Coupled Networks ◽  
Coupled Maps ◽  
Period Doubling Bifurcations

Network architecture can lead to robust synchrony in coupled maps and to codimension one bifurcations from synchronous fixed-points at which the associated Jacobian is nilpotent. We discuss the codimension one synchrony-breaking period-doubling bifurcations for three-cell coupled maps. Interesting phenomena occur for all these coupled maps — a branch of period-2 points with amplitude growing as |λ|⅙ for coupled networks of feed-forward type, as well as multiple (two) branches of period-2 points with amplitude growing as |λ|½ for coupled networks of feed-forward type. We also discuss how some results related to patterns of synchrony that are valid for coupled vector fields are also valid for coupled maps.

FORCED SYNCHRONIZATION IN MORRIS–LECAR NEURONS

2007 ◽  
Vol 17 (10) ◽  
pp. 3523-3528 ◽  
Author(s):  
HIROYUKI KITAJIMA ◽  
JÜRGEN KURTHS
Keyword(s):  
Bifurcation Analysis ◽  
Coefficient Of Variation ◽  
Interspike Interval ◽  
Period Doubling ◽  
Class Ii ◽  
Bifurcation Structure ◽  
Periodic Oscillations ◽  
Numerical Bifurcation ◽  
Period Doubling Bifurcations ◽  
Numerical Bifurcation Analysis

We investigate forced synchronization between electrically coupled Morris–Lecar neurons with class I and class II excitability through numerical bifurcation analysis. We find that class II neurons have wider parameter regions of forced synchronization. However, the bifurcation structure and patterns of spikes for class II are complicated; there exist period-doubling bifurcations, interesting two-periodic oscillations and irregular bursting spikes with high values of the coefficient of variation of the interspike interval.

THE BIFURCATION STRUCTURE OF FIRING PATTERN TRANSITIONS IN THE CHAY NEURONAL PACEMAKER MODEL

2008 ◽  
Vol 16 (01) ◽  
pp. 33-49 ◽  
Author(s):  
ZHUOQIN YANG ◽  
QISHAO LU
Keyword(s):  
Bifurcation Analysis ◽  
Calcium Concentration ◽  
Period Doubling ◽  
Nerve Fibers ◽  
Neuronal Firing ◽  
Dynamical Parameter ◽  
True Nature ◽  
Firing Patterns ◽  
Chaotic Bursting ◽  
Period Doubling Bifurcations

Different transitions of neuronal firing patterns are explored by the combination of experimental results, numerical simulation and bifurcation analysis. Three types of firing sequences with respect to extracellular calcium concentration ([ Ca 2+] o ) were observed in experiments on neural pacemakers. In accordance with them, the corresponding transitions of neuronal firing patterns are surveyed by standard bifurcation analysis of the Chay model, where λn corresponds to different nerve fibers and VC is the dynamical parameter. The results are listed in this paper. Firstly, it is obtained that the transitions of periodic firing patterns from period-1 bursting to period-1 spiking without any bifurcation, from period-1 to -2 to -1 through a pair of period-doubling bifurcations and from period-1 to -2 to -1 through two pairs of period-doubling bifurcations. Secondly, one supercritical and two subcritical period-doubling bursting sequences with different appearances lead to chaos, respectively. Then the former transits directly to an inverse supercritical period-doubling spiking sequence via chaos, and the latter transit to it through the period-adding bursting sequences from period-1 to -3 and from -1 to -5 with chaotic bursting, respectively. Thirdly, we reveal the true nature of period-adding bursting sequence without chaotic bursting. Every periodic bursting is closely related to two period-doubling bifurcations of the corresponding periodic spiking, except for period-1 bursting appearing via Hopf bifurcation and disappearing via period-doubling bifurcation. As a consequence, the period-adding bursting sequence without chaotic bursting has a compound structure of elementary bifurcations with transitions from spiking to bursting. Thus period-adding bifurcation without chaos cannot be regarded as a new elementary bifurcation.

Boundary-Crisis-Induced Complex Bursting Patterns in a Forced Cubic Map

2017 ◽  
Vol 27 (04) ◽  
pp. 1750051 ◽  
Author(s):  
Xiujing Han ◽  
Chun Zhang ◽  
Yue Yu ◽  
Qinsheng Bi
Keyword(s):  
Fixed Points ◽  
Periodic Orbits ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Period 2 ◽  
Boundary Crisis ◽  
Underlying Mechanisms ◽  
Parameter Values ◽  
Point Type ◽  
Period Doubling Bifurcations

This paper reports novel routes to complex bursting patterns based on a forced cubic map, in which boundary-crisis-induced novel bursting patterns are investigated. Typically, the cubic map exhibits stable upper and lower branches of fixed points, which may evolve into chaos in opposite parameter directions by a cascade of period-doubling bifurcations. We show that the chaotic attractors on the stable branches may suddenly disappear by boundary crisis, thus leading to fast transitions from chaos to other attractors and giving rise to switchings between the stable branches of solutions of the cubic map. In particular, the attractors that the trajectory switches to by boundary crisis can be fixed points, periodic orbits and chaos, dependent on parameter values of the cubic map, and this helps us to reveal three general types of boundary-crisis-induced bursting, i.e. bursting of chaos-point type, bursting of chaos-cycle type and bursting of chaos-chaos type. Moreover, each bursting type may contain various bursting patterns. For bursting of chaos-cycle type, we see rich bursting patterns, e.g. chaos-period-2 bursting, chaos-period-4 bursting, chaos-period-8 bursting, etc. Our results enrich the possible routes to complex bursting patterns as well as the underlying mechanisms of complex bursting patterns.

Path-Following Bifurcation Analysis of Church Bell Dynamics

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Antonio Simon Chong Escobar ◽  
Piotr Brzeski ◽  
Marian Wiercigroch ◽  
Przemyslaw Perlikowski
Keyword(s):  
Bifurcation Analysis ◽  
Applied Mathematics ◽  
Path Following ◽  
Engineering Society ◽  
Effective Performance ◽  
Computational Science And Engineering ◽  
Novel Method ◽  
Direct Numerical Integration ◽  
Path Following Methods ◽  
Church Bell

In this paper, we perform a path-following bifurcation analysis of church bell to gain an insight into the governing dynamics of the yoke–bell–clapper system. We use an experimentally validated hybrid dynamical model based on the detailed measurements of a real church bell. Numerical analysis is performed both by a direct numerical integration and a path-following methods using a new numerical toolbox ABESPOL (Chong, 2016, “Numerical Modeling and Stability Analysis of Non-Smooth Dynamical Systems Via ABESPOL,” Ph.D. thesis, University of Aberdeen, Aberdeen, UK) based on COCO (Dankowicz and Schilder, Recipes for Continuation (Computational Science and Engineering), Society for Industrial and Applied Mathematics, Philadelphia, PA). We constructed one-parameter diagrams that allow to characterize the most common dynamical states and to investigate the mechanisms of their dynamic stability. A novel method allowing to locate the regions in the parameters' space ensuring robustness of bells' effective performance is presented.

Period-multiplying cascades for diffeomorphisms of the disc

1994 ◽  
Vol 116 (2) ◽  
pp. 359-374 ◽  
Author(s):  
Jean-Marc Gambaudo ◽  
John Guaschi ◽  
Toby Hall
Keyword(s):  
Fixed Point ◽  
Periodic Orbits ◽  
Analogous Result ◽  
Period Doubling ◽  
Positive Entropy ◽  
One Dimensional ◽  
Continuous Map ◽  
Period 2 ◽  
Zero Entropy ◽  
Period Doubling Bifurcations

It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

scholarly journals ANALYSIS OF BIFURCATIONS IN A POWER SYSTEM MODEL WITH EXCITATION LIMITS

2001 ◽  
Vol 11 (09) ◽  
pp. 2509-2516 ◽  
Author(s):  
RAJESH G. KAVASSERI ◽  
K. R. PADIYAR
Keyword(s):  
Power System ◽  
Bifurcation Analysis ◽  
Smooth Function ◽  
Period Doubling ◽  
System Model ◽  
System Behavior ◽  
Multiple Attractors ◽  
Period Doubling Bifurcations ◽  
Power System Model ◽  
Qualitative Changes

This paper studies bifurcations in a three node power system when excitation limits are considered. This is done by approximating the limiter by a smooth function to facilitate bifurcation analysis. Spectacular qualitative changes in the system behavior induced by the limiter are illustrated by two case studies. Period doubling bifurcations and multiple attractors are shown to result due to the limiter. Detailed numerical simulations are presented to verify the results and illustrate the nature of the attractors and solutions involved.

scholarly journals DYNAMICS OF A SINGLE MODEL NEURON

1993 ◽  
Vol 03 (02) ◽  
pp. 271-278 ◽  
Author(s):  
FRANK PASEMANN
Keyword(s):  
Model Neuron ◽  
Period Doubling ◽  
Underlying Mechanism ◽  
Cusp Catastrophe ◽  
Stationary States ◽  
Single Model ◽  
Excitatory Connection ◽  
Stable Period ◽  
Period 2 ◽  
Period Doubling Bifurcations

The parametrized dynamics of a standard nonlinear model neuron with self-interaction is discussed. For units with a self-excitatory connection a hysteresis effect is observed, and the underlying mechanism is identified as that of a cusp catastrophe. This is true for discrete as well as for continuous dynamics. For the discrete dynamics of self-inhibiting units there appear period-doubling bifurcations from stationary states to stable period-2 orbits.

scholarly journals Zero Average Surface Controlled Boost-Flyback Converter

Energies ◽  
2020 ◽  
Vol 14 (1) ◽  
pp. 57
Author(s):  
Juan-Guillermo Muñoz ◽  
Fabiola Angulo ◽  
David Angulo-Garcia
Keyword(s):  
Duty Cycle ◽  
Period Doubling ◽  
Power Converter ◽  
Control Technique ◽  
Average Surface ◽  
Flyback Converter ◽  
Stable Period ◽  
Period 2 ◽  
Wide Range ◽  
The Right

The boost-flyback converter is a DC-DC step-up power converter with a wide range of technological applications. In this paper, we analyze the boost-flyback dynamics when controlled via a modified Zero-Average-Dynamics control technique, hereby named Zero-Average-Surface (ZAS). While using the ZAS strategy, it is possible to calculate the duty cycle at each PWM cycle that guarantees a desired stable period-1 solution, by forcing the system to evolve in such way that a function that is constructed with strategical combination of the states over the PWM period has a zero average. We show, by means of bifurcation diagrams, that the period-1 orbit coexists with a stable period-2 orbit with a saturated duty cycle. While using linear stability analysis, we demonstrate that the period-1 orbit is stable over a wide range of parameters and it loses stability at high gains and low loads via a period doubling bifurcation. Finally, we show that, under the right choice of parameters, the period-1 orbit controller with ZAS strategy satisfactorily rejects a wide range of disturbances.

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