ON THE DYNAMICS OF A SIMPLE RATIONAL PLANAR MAP

2013 ◽  
Vol 23 (06) ◽  
pp. 1330021
Author(s):  
CHRISTOFOROS SOMARAKIS ◽  
JOHN S. BARAS
Keyword(s):  
Algebraic Structure ◽  
Period Doubling ◽  
Strange Attractors ◽  
Nonlinear Phenomena ◽  
Bifurcation Diagrams ◽  
Planar Map ◽  
Period Doubling Bifurcations

The dynamics of the map [Formula: see text] are discussed for various values of its parameters. Despite the simple algebraic structure, this map, recently introduced in the literature, is very rich in nonlinear phenomena. Multiple strange attractors, transitions to chaos via period-doubling bifurcations, quasiperiodicity as well as intermittency, interior crisis, hyperchaos are only a few. In this work, strange attractors, bifurcation diagrams, periodic windows, invariant characteristics are investigated both analytically and numerically.

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.

scholarly journals Predicting Tipping Points in Chaotic Maps with Period-Doubling Bifurcations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Changzhi Li ◽  
Dhanagopal Ramachandran ◽  
Karthikeyan Rajagopal ◽  
Sajad Jafari ◽  
Yongjian Liu
Keyword(s):  
Chaotic Maps ◽  
Period Doubling ◽  
Tipping Points ◽  
Bifurcation Parameter ◽  
Bifurcation Diagrams ◽  
Bifurcation Points ◽  
Slowing Down ◽  
Sine Map ◽  
Autocorrelation Method ◽  
Period Doubling Bifurcations

In this paper, bifurcation points of two chaotic maps are studied: symmetric sine map and Gaussian map. Investigating the properties of these maps shows that they have a variety of dynamical solutions by changing the bifurcation parameter. Sine map has symmetry with respect to the origin, which causes multistability in its dynamics. The systems’ bifurcation diagrams show various dynamics and bifurcation points. Predicting bifurcation points of dynamical systems is vital. Any bifurcation can cause a huge wanted/unwanted change in the states of a system. Thus, their predictions are essential in order to be prepared for the changes. Here, the systems’ bifurcations are studied using three indicators of critical slowing down: modified autocorrelation method, modified variance method, and Lyapunov exponent. The results present the efficiency of these indicators in predicting bifurcation points.

CHAOTIC DYNAMICS OF A FIVE-DIMENSIONAL NONLINEAR NETWORK

2007 ◽  
Vol 18 (03) ◽  
pp. 335-342
Author(s):  
XUEWEI JIANG ◽  
DI YUAN ◽  
YI XIAO
Keyword(s):  
Lyapunov Exponent ◽  
Power Spectrum ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Chinese Traditional Medicine ◽  
Bifurcation Diagrams ◽  
Nonlinear Network ◽  
Dynamical Behaviors ◽  
Lyapunov Exponent Spectrum ◽  
Period Doubling Bifurcations

The dynamics of a five-dimensional nonlinear network based on the theory of Chinese traditional medicine is studied by the Lyapunov exponent spectrum, Poincaré, power spectrum and bifurcation diagrams. The result shows that this system has complex dynamical behaviors, such as chaotic ones. It also shows that the system evolves into chaos through a series of period-doubling bifurcations.

scholarly journals Asymmetric Double Strange Attractors in a Simple Autonomous Jerk Circuit

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
G. H. Kom ◽  
J. Kengne ◽  
J. R. Mboupda Pone ◽  
G. Kenne ◽  
A. B. Tiedeu
Keyword(s):  
Autonomous System ◽  
Period Doubling ◽  
Bifurcation Diagrams ◽  
Colpitts Oscillator ◽  
Exponential Nonlinearity ◽  
Unusual Phenomenon ◽  
Initial States ◽  
Type Of Behavior ◽  
Period Doubling Bifurcations ◽  
Lower Dimensional

The dynamics of a simple autonomous jerk circuit previously introduced by Sprott in 2011 are investigated. In this paper, the model is described by a three-time continuous dimensional autonomous system with an exponential nonlinearity. Using standard nonlinear techniques such as time series, bifurcation diagrams, Lyapunov exponent plots, and Poincaré sections, the dynamics of the system are characterized with respect to its parameters. Period-doubling bifurcations, periodic windows, and coexisting bifurcations are reported. As a major result of this work, it is found that the system experiences the unusual phenomenon of asymmetric bistability marked by the presence of two different attractors (e.g., screw-like Shilnikov attractor with a spiralling-like Feigenbaum attractor) for the same parameters setting, depending solely on the choice of initial states. Among few cases of lower dimensional systems capable of such type of behavior reported to date (e.g., Colpitts oscillator, Newton–Leipnik system, and hyperchaotic oscillator with gyrators), the jerk circuit/system considered in this work represents the simplest prototype. Results of theoretical analysis are perfectly reproduced by laboratory experimental measurements.

Cantilever Dynamics With Attractive and Repulsive Tip Interactions

2009 ◽  
Author(s):  
Ishita Chakraborty ◽  
Balakumar Balachandran
Keyword(s):  
Period Doubling ◽  
Experimental System ◽  
Nonlinear Phenomena ◽  
Strong Nonlinearity ◽  
Force Microscopy ◽  
Compliant Surface ◽  
Atomic Force ◽  
Macro Scale ◽  
Repulsive Forces ◽  
Period Doubling Bifurcations

In tapping mode atomic force microscopy (AFM) operations, typically, a micro-scale cantilever undergoes long-range attractive and short-range repulsive forces as it approaches the surface. Due to the strong nonlinearity of the tip-sample interaction forces, a variety of nonlinear phenomena has been observed. In order to obtain a better understanding of such phenomena, in this work, a macro-scale experimental system is constructed with attractive and repulsive tip interactions. Magnetic forces are used to generate the attractive forces, and impact with a compliant surface is used to produce the repulsive forces. A reduced-order model for this system has been developed with Derjaguin-Muller-Toporov (DMT) contact mechanics. In prior work conducted in the group, period-doubling bifurcations have been observed close to near grazing conditions when AFM cantilevers are operated between first and second natural frequencies of the system. This phenomenon is further explored in this article.

The Croonian Lecture, 1985 - When two and two do not make four: nonlinear phenomena in ecology

1986 ◽  
Vol 228 (1252) ◽  
pp. 241-266 ◽  
Keyword(s):  
Natural Populations ◽  
Period Doubling ◽  
Dynamical Behaviour ◽  
Nonlinear Phenomena ◽  
Deterministic Models ◽  
Deterministic Systems ◽  
Management Of Resources ◽  
Random Fluctuations ◽  
Stable Points ◽  
Period Doubling Bifurcations

The simplest mathematical models describing the dynamics of natural populations of plants and animals are nonlinear. These models can exhibit an astonishing array of dynamical behaviour, ranging from stable points to period-doubling bifurcations that produce a cascade of stable cycles, to apparently random fluctuations; that is, simple deterministic systems can produce chaotic dynamics. This review shows how these ideas illuminate some of the observed properties of real populations in the field and laboratory, and explores some of the practical implications. When unpredictable environmental fluctuations are superimposed on such deterministic models, there are further complications both in the analysis and interpretation of data (what factors regulate the population ?) and in the management of resources (how should fish quotas be set in an uncertain environment ?).

scholarly journals Controlling Chaos through Period-Doubling Bifurcations in Attitude Dynamics for Power Systems

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Shun-Chang Chang
Keyword(s):  
Power Systems ◽  
Control Method ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Lyapunov Stability Theory ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Frequency Spectra ◽  
Controlling Chaos ◽  
Period Doubling Bifurcations

This paper addresses the complex nonlinear dynamics involved in controlling chaos in power systems using bifurcation diagrams, time responses, phase portraits, Poincaré maps, and frequency spectra. Our results revealed that nonlinearities in power systems produce period-doubling bifurcations, which can lead to chaotic motion. Analysis based on the Lyapunov exponent and Lyapunov dimension was used to identify the onset of chaotic behavior. We also developed a continuous feedback control method based on synchronization characteristics for suppressing of chaotic oscillations. The results of our simulation support the feasibility of using the proposed method. The robustness of parametric perturbations on a power system with synchronization control was analyzed using bifurcation diagrams and Lyapunov stability theory.

Chaotic Vibration and Internal Resonance Phenomena in Rotor Systems

2005 ◽  
Vol 128 (2) ◽  
pp. 156-169 ◽  
Author(s):  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Internal Resonance ◽  
Critical Speed ◽  
Period Doubling ◽  
Nonlinear Phenomena ◽  
Rotor Systems ◽  
Chaotic Vibration ◽  
Consecutive Period ◽  
Route To Chaos ◽  
Harmonic Resonance ◽  
Period Doubling Bifurcations

Rotating machinery has effects of gyroscopic moments, but most of them are small. Then, many kinds of rotor systems satisfy the relation of 1 to (−1) type internal resonance approximately. In this paper, the dynamic characteristics of nonlinear phenomena, especially chaotic vibration, due to the 1 to (−1) type internal resonance at the major critical speed and twice the major critical speed are investigated. The following are clarified theoretically and experimentally: (a) the Hopf bifurcation and consecutive period doubling bifurcations possible route to chaos occur from harmonic resonance at the major critical speed and from subharmonic resonance at twice the major critical speed, (b) another chaotic vibration from the combination resonance occurs at twice the major critical speed. The results demonstrate that chaotic vibration may occur even in the rotor system with weak nonlinearity when the effect of the gyroscopic moment is small.

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.

Some Nonlinear Effects of Magnetic Bearings

1999 ◽  
Author(s):  
Norbert Steinschaden ◽  
Helmut Springer
Keyword(s):  
Equations Of Motion ◽  
Period Doubling ◽  
Path Following ◽  
Nonlinear Effects ◽  
Magnetic Bearing ◽  
Operating Conditions ◽  
Active Magnetic Bearing ◽  
Nonlinear Phenomena ◽  
Amb System ◽  
Large Rotor

Abstract In order to get a better understanding of the dynamics of active magnetic bearing (AMB) systems under extreme operating conditions a simple, nonlinear model for a radial AMB system is investigated. Instead of the common way of linearizing the magnetic forces at the center position of the rotor with respect to rotor displacement and coil current, the fully nonlinear force to displacement and the force to current characteristics are used. The AMB system is excited by unbalance forces of the rotor. Especially for the case of large rotor eccentricities, causing large rotor displacements, the behaviour of the system is discussed. A path-following analysis of the equations of motion shows that for some combinations of parameters well-known nonlinear phenomena may occur, as, for example, symmetry breaking, period doubling and even regions of global instability can be observed.

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