scholarly journals Another New Chaotic System: Bifurcation and Chaos Control

2020 ◽  
Vol 30 (11) ◽  
pp. 2050161
Author(s):  
Arnob Ray ◽  
Dibakar Ghosh
Keyword(s):  
Hopf Bifurcation ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Period Doubling ◽  
Slow Manifold ◽  
Singular Perturbation Theory ◽  
Geometric Singular Perturbation Theory ◽  
Dimensional Parameter ◽  
Lyapunov Coefficient ◽  
The Stability

We propose a new simple three-dimensional continuous autonomous model with two nonlinear terms and observe the dynamical behavior with respect to system parameters. This system changes the stability of fixed point via Hopf bifurcation and then undergoes a cascade of period-doubling route to chaos. We analytically derive the first Lyapunov coefficient to investigate the nature of Hopf bifurcation. We investigate well-separated regions for different kinds of attractors in the two-dimensional parameter space. Next, we introduce a timescale ratio parameter and calculate the slow manifold using geometric singular perturbation theory. Finally, the chaotic state annihilates by decreasing the value of the timescale ratio parameter.

scholarly journals Stability and Bifurcation Analysis of a Modified Epidemic Model for Computer Viruses

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Chuandong Li ◽  
Wenfeng Hu ◽  
Tingwen Huang
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Normal Form Theory ◽  
Computer Viruses ◽  
Stability And Bifurcation ◽  
Form Theory ◽  
The Stability

We extend the three-dimensional SIR model to four-dimensional case and then analyze its dynamical behavior including stability and bifurcation. It is shown that the new model makes a significant improvement to the epidemic model for computer viruses, which is more reasonable than the most existing SIR models. Furthermore, we investigate the stability of the possible equilibrium point and the existence of the Hopf bifurcation with respect to the delay. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. An analytical condition for determining the direction, stability, and other properties of bifurcating periodic solutions is obtained by using the normal form theory and center manifold argument. The obtained results may provide a theoretical foundation to understand the spread of computer viruses and then to minimize virus risks.

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

Turret Mooring Design Based on Analytical Expressions of Catastrophes of Slow-Motion Dynamics

1998 ◽  
Vol 120 (3) ◽  
pp. 154-164 ◽  
Author(s):  
M. M. Bernitsas ◽  
L. O. Garza-Rios
Keyword(s):  
Parametric Design ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Slow Motion ◽  
Sensitivity Analyses ◽  
Design Parameters ◽  
Mooring Lines ◽  
The Stability ◽  
Fifth Order ◽  
Analytical Expressions

Analytical expressions of the bifurcation boundaries exhibited by turret mooring systems (TMS), and expressions that define the morphogeneses occurring across boundaries are developed. These expressions provide the necessary means for evaluating the stability of a TMS around an equilibrium position, and constructing catastrophe sets in two or three-dimensional parametric design spaces. Sensitivity analyses of the bifurcation boundaries define the effect of any parameter or group of parameters on the dynamical behavior of the system. These expressions allow the designer to select appropriate values for TMS design parameters without resorting to trial and error. A four-line TMS is used to demonstrate this design methodology. The mathematical model consists of the nonlinear, fifth-order, low-speed, large-drift maneuvering equations. Mooring lines are modeled with submerged catenaries, and include nonlinear drag. External excitation consists of time-independent current, wind, and mean wave drift.

Graphical Hopf Bifurcation of a Filippov HTLV-1 Model With Delay in Cytotoxic T Cells Response

2018 ◽  
Vol 140 (9) ◽  
Author(s):  
Elham Shamsara ◽  
Zahra Afsharnezhad ◽  
Elham Javidmanesh
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Bifurcation Theory ◽  
Cytotoxic T Cells ◽  
Dynamical Behavior ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach

In this paper, we present a discontinuous cytotoxic T cells (CTLs) response for HTLV-1. Moreover, a delay parameter for the activation of CTLs is considered. In fact, a system of differential equation with discontinuous right-hand side with delay is defined for HTLV-1. For analyzing the dynamical behavior of the system, graphical Hopf bifurcation is used. In general, Hopf bifurcation theory will help to obtain the periodic solutions of a system as parameter varies. Therefore, by applying the frequency domain approach and analyzing the associated characteristic equation, the existence of Hopf bifurcation by using delay immune response as a bifurcation parameter is determined. The stability of Hopf bifurcation periodic solutions is obtained by the Nyquist criterion and the graphical Hopf bifurcation theorem. At the end, numerical simulations demonstrated our results for the system of HTLV-1.

Dynamics and Stability of Partial Immersion Milling Operations

1999 ◽  
Author(s):  
M. X. Zhao ◽  
B. Balachandran ◽  
M. A. Davies ◽  
J. R. Pratt
Keyword(s):  
Hopf Bifurcation ◽  
Time Variation ◽  
Periodic Motion ◽  
Contact Time ◽  
Period Doubling ◽  
Experimental Investigations ◽  
Partial Immersion ◽  
Milling Operations ◽  
Wide Range ◽  
The Stability

Abstract In this paper, numerical and experimental investigations conducted into the dynamics and stability of partial immersion milling operations are presented. A mechanics based model is used for simulations of a wide range of milling operations and instabilities that arise due to regeneration and/or impact effects are studied. Poincaré sections are used to assess the stability of motions. The studies reveal that apart from Hopf bifurcation of a periodic motion, a period-doubling bifurcation of a periodic motion may also lead to chatter in partial immersion milling operations. Issues such as tooth contact time variation and structure of stability charts are also discussed.

scholarly journals Numerical Exploration of Kaldorian Interregional Macrodynamics: Enhanced Stability and Predominance of Period Doubling under Flexible Exchange Rates

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
Toichiro Asada ◽  
Christos Douskos ◽  
Vassilis Kalantonis ◽  
Panagiotis Markellos
Keyword(s):  
Exchange Rates ◽  
Government Expenditure ◽  
Period Doubling ◽  
Numerical Approach ◽  
Capital Movement ◽  
Dimensional Parameter ◽  
Stability Of Equilibrium ◽  
The Stability ◽  
Flexible Exchange Rates ◽  
Enhanced Stability

We present a discrete two-regional Kaldorian macrodynamic model with flexible exchange rates and explore numerically the stability of equilibrium and the possibility of generation of business cycles. We use a grid search method in two-dimensional parameter subspaces, and coefficient criteria for the flip and Hopf bifurcation curves, to determine the stability region and its boundary curves in several parameter ranges. The model is characterized by enhanced stability of equilibrium, while its predominant asymptotic behavior when equilibrium is unstable is period doubling. Cycles are scarce and short-lived in parameter space, occurring at large values of the degree of capital movementβ. By contrast to the corresponding fixed exchange rates system, for cycles to occur sufficient amount of trade is requiredtogetherwith high levels of capital movement. Rapid changes in exchange rate expectations and decreased government expenditure are factors contributing to the creation of interregional cycles. Examples of bifurcation and Lyapunov exponent diagrams illustrating period doubling or cycles, and their development into chaotic attractors, are given. The paper illustrates the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters.

scholarly journals Stability and Bifurcation Analysis of a Three-Dimensional Recurrent Neural Network with Time Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Yingguo Li
Keyword(s):  
Neural Network ◽  
Time Delay ◽  
Recurrent Neural Network ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Three Dimensional ◽  
Nonlinear Dynamical ◽  
Bifurcation Direction ◽  
The Stability ◽  
Manifold Theory

We consider the nonlinear dynamical behavior of a three-dimensional recurrent neural network with time delay. By choosing the time delay as a bifurcation parameter, we prove that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Applying the nor- mal form method and center manifold theory, we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution. Some numerical examples are also presented to verify the theoretical analysis.

scholarly journals Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
G. Kai ◽  
W. Zhang ◽  
Z. C. Wei ◽  
J. F. Wang ◽  
A. Akgul
Keyword(s):  
Hopf Bifurcation ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Financial System ◽  
Invariant Set ◽  
Phase Portraits ◽  
Lyapunov Coefficient ◽  
Stable Periodic ◽  
Analytical Proof ◽  
Positively Invariant Set

This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.

Antimonotonicity, Chaos and Multiple Attractors in a Novel Autonomous Jerk Circuit

2017 ◽  
Vol 27 (07) ◽  
pp. 1750100 ◽  
Author(s):  
J. Kengne ◽  
A. Nguomkam Negou ◽  
Z. T. Njitacke
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Electrical Circuit ◽  
Semiconductor Diode ◽  
Chaotic Attractors ◽  
Coexisting Attractors ◽  
Multiple Attractors ◽  
Systematic Analysis ◽  
The Stability

We perform a systematic analysis of a system consisting of a novel jerk circuit obtained by replacing the single semiconductor diode of the original jerk circuit described in [Sprott, 2011a] with a pair of semiconductor diodes connected in antiparallel. The model is described by a continuous time three-dimensional autonomous system with hyperbolic sine nonlinearity, and may be viewed as a control system with nonlinear velocity feedback. The stability of the (unique) fixed point, the local bifurcations, and the discrete symmetries of the model equations are discussed. The complex behavior of the system is categorized in terms of its parameters by using bifurcation diagrams, Lyapunov exponents, time series, Poincaré sections, and basins of attraction. Antimonotonicity, period doubling bifurcation, symmetry restoring crises, chaos, and coexisting bifurcations are reported. More interestingly, one of the key contributions of this work is the finding of various regions in the parameters’ space in which the proposed (“elegant”) jerk circuit experiences the unusual phenomenon of multiple competing attractors (i.e. coexistence of four disconnected periodic and chaotic attractors). The basins of attraction of various coexisting attractors display complexity (i.e. fractal basins boundaries), thus suggesting possible jumps between coexisting attractors in experiment. Results of theoretical analyses are perfectly traced by laboratory experimental measurements. To the best of the authors’ knowledge, the jerk circuit/system introduced in this work represents the simplest electrical circuit (only a quadruple op amplifier chip without any analog multiplier chip) reported to date capable of four disconnected periodic and chaotic attractors for the same parameters setting.

On transition of the pulsatile pipe flow

1986 ◽  
Vol 170 ◽  
pp. 169-197 ◽  
Author(s):  
J. C. Stettler ◽  
A. K. M. Fazle Hussain
Keyword(s):  
Steady Flow ◽  
Pipe Flow ◽  
Three Dimensional ◽  
Building Blocks ◽  
Rate Of Change ◽  
Turbulent Pipe Flow ◽  
Dimensional Parameter ◽  
Pipe Length ◽  
The Mean ◽  
The Stability

Transition in a pipe flow with a superimposed sinusoidal modulation has been studied in a straight circular water pipe using laser-Doppler anemometer (LDA) techniques. This study has determined the stability–transition boundary in the three-dimensional parameter space defined by the mean and modulation Reynolds numbers Rem, Remω and the frequency parameter λ. Furthermore, it documents the mean passage frequency Fp of ‘turbulent plugs’ as functions of Rem’ Remω and λ. This study also delineates the conditions when plugs occur randomly in time (as in the steady flow) or phase-locked with the excitation. The periodic flow requires a new definition of the transitional Reynolds number Rer, identified on the basis of the rate of change of Fp with Rem. The extent of increase or decrease in Rer from the corresponding steady flow value depends on λ and Remω. At any Rem and Remω, maximum stabilization occurs at λ ≈ 5. With increasing Remω, the ‘stabilization bandwidth’ of modulation frequencies increases and then abruptly decreases after levelling off. The maximum stabilization bandwidth depends strongly on Rem, decreasing with increasing Rem. Previously reported observations of turbulence during deceleration, followed by a relaminarization during acceleration, can be explained in terms of a new phenomenon: namely, periodic modulation produces longitudinally periodic cells of turbulent fluid ‘plugs’ which differ in structural details from ‘puffs’ or ‘slugs’ in steady transitional pipe flows and are called patches. The length of a patch could be increased continuously from zero to the entire pipe length by increasing Rem. This tends to question the concept that all turbulent plugs (and even the fully-turbulent pipe flow) consists of many identical elementary plugs as basic ‘building blocks’.

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