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.

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.

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.

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 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.

scholarly journals Can short time delays influence the variability of the solar cycle?

2010 ◽  
Vol 6 (S271) ◽  
pp. 288-296
Author(s):  
Laurène Jouve ◽  
Michael R. E. Proctor ◽  
Geoffroy Lesur
Keyword(s):  
Solar Cycle ◽  
Convection Zone ◽  
Mean Field ◽  
Period Doubling ◽  
Temporal Modulation ◽  
Solar Convection Zone ◽  
Flux Tubes ◽  
Solar Convection ◽  
Short Time ◽  
Period Doubling Bifurcations

AbstractWe present the effects of introducing results of 3D MHD simulations of buoyant magnetic fields in the solar convection zone in 2D mean-field Babcock-Leighton models. In particular, we take into account the time delay introduced by the rise time of the toroidal structures from the base of the convection zone to the solar surface. We find that the delays produce large temporal modulation of the cycle amplitude even when strong and thus rapidly rising flux tubes are considered. The study of a reduced model reveals that aperiodic modulations of the solar cycle appear after a sequence of period doubling bifurcations typical of non-linear systems. We also discuss the memory of such systems and the conclusions which may be drawn concerning the actual solar cycle variability.

Bifurcations and Chaos in a Minimal Neural Network: Forced Coupled Neurons

2021 ◽  
Vol 31 (10) ◽  
pp. 2150147
Author(s):  
Yo Horikawa
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Output Function ◽  
Piecewise Constant ◽  
Border Collision Bifurcations ◽  
Saddle Node ◽  
Periodic Input ◽  
Symmetric Periodic Solution ◽  
Period Doubling Bifurcations

The bifurcations and chaos in a system of two coupled sigmoidal neurons with periodic input are revisited. The system has no self-coupling and no inherent limit cycles in contrast to the previous studies and shows simple bifurcations qualitatively different from the previous results. A symmetric periodic solution generated by the periodic input underdoes a pitchfork bifurcation so that a pair of asymmetric periodic solutions is generated. A chaotic attractor is generated through a cascade of period-doubling bifurcations of the asymmetric periodic solutions. However, a symmetric periodic solution repeats saddle-node bifurcations many times and the bifurcations of periodic solutions become complicated as the output gain of neurons is increasing. Then, the analysis of border collision bifurcations is carried out by using a piecewise constant output function of neurons and a rectangular wave as periodic input. The saddle-node, the pitchfork and the period-doubling bifurcations in the coupled sigmoidal neurons are replaced by various kinds of border collision bifurcations in the coupled piecewise constant neurons. Qualitatively the same structure of the bifurcations of periodic solutions in the coupled sigmoidal neurons is derived analytically. Further, it is shown that another period-doubling route to chaos exists when the output function of neurons is asymmetric.

Nonstationary Response of a Two-Degrees-of-Freedom Nonlinear Ship Model Under Modulated Excitation

1995 ◽  
Author(s):  
Ruigui Pan ◽  
Huw G. Davies
Keyword(s):  
Degrees Of Freedom ◽  
Stability Boundary ◽  
Period Doubling ◽  
Approximate Solutions ◽  
Loss Of Stability ◽  
Two Degrees Of Freedom ◽  
Ship Model ◽  
Nonstationary Response ◽  
Period 2 ◽  
The Stability

Abstract Nonstationary response of a two-degrees-of-freedom system with quadratic coupling under a time varying modulated amplitude sinusoidal excitation is studied. The nonlinearly coupled pitch and roll ship model is based on Nayfeh, Mook and Marshall’s work for the case of stationary excitation. The ship model has a 2:1 internal resonance and is excited near the resonance of the pitch mode. The modulated excitation (F0 + F1 cos ωt) cosQt is used to model a narrow band sea-wave excitation. The response demonstrates a variety of bifurcations, loss of stability, and chaos phenomena that are not present in the stationary case. We consider here the periodically modulated response. Chaotic response of the system is discussed in a separate paper. Several approximate solutions, under both small and large modulating amplitudes F1, are obtained and compared with the exact one. The stability of an exact solution with one mode having zero amplitude is studied. Loss of stability in this case involves either a rapid transition from one of two stable (in the stationary sense) branches to another, or a period doubling bifurcation. From Floquet theory, various stability boundary diagrams are obtained in F1 and F0 parameter space which can be used to predict the various transition phenomena and the period-2 bifurcations. The study shows that both the modulation parameters F1 and ω (the modulating frequency) have great effect on the stability boundaries. Because of the modulation, the stable area is greatly expanded, and the stationary bifurcation point can be exceeded without loss of stability. Decreasing ω can make the stability boundary very complicated. For very small ω the response can make periodic transitions between the two (pseudo) stable solutions.

Two Groups of Chaotic Vibrations in a Single-Degree-of-Freedom Maglev System

1997 ◽  
Author(s):  
Zhixiang Xu ◽  
Hideyuki Tamura
Keyword(s):  
Magnetic Levitation ◽  
Excitation Frequency ◽  
Power Spectra ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Degree Of Freedom ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Moving Base ◽  
Period Doubling Bifurcations

Abstract In this paper, a single-degree-of-freedom magnetic levitation dynamic system, whose spring is composed of a magnetic repulsive force, is numerically analyzed. The numerical results indicate that a body levitated by magnetic force shows many kinds of vibrations upon adjusting the system parameters (viz., damping, excitation amplitude and excitation frequency) when the system is excited by the harmonically moving base. For a suitable combination of parameters, an aperiodic vibration occurs after a sequence of period-doubling bifurcations. Typical aperiodic vibrations that occurred after period-doubling bifurcations from several initial states are identified as chaotic vibration and classified into two groups by examining their power spectra, Poincare maps, fractal dimension analyses, etc.

scholarly journals Chaotic dynamics of a pulse modulated control system for a heating unit

2018 ◽  
Vol 224 ◽  
pp. 02055
Author(s):  
Yuriy A. Gol’tsov ◽  
Alexander S. Kizhuk ◽  
Vasiliy G. Rubanov
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classical Period ◽  
Nonlinear Phenomena ◽  
Pulse Control ◽  
Heating Unit ◽  
Border Collision Bifurcation ◽  
Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

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