STUDY OF DEGENERATE BIFURCATIONS IN MAPS: A FEEDBACK SYSTEMS APPROACH

2004 ◽  
Vol 14 (05) ◽  
pp. 1625-1641 ◽  
Author(s):  
MARÍA BELÉN D'AMICO ◽  
JORGE L. MOIOLA ◽  
EDUARDO E. PAOLINI
Keyword(s):  
Discrete Time ◽  
Harmonic Balance ◽  
Systems Approach ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Hopf Bifurcations ◽  
Feedback Systems ◽  
Nonlinear Maps ◽  
Frequency Domain Approach ◽  
Order Extension

The dynamical behavior of nonlinear maps undergoing degenerate period doubling or degenerate Hopf bifurcations is studied via a frequency-domain approach. The technique is based on a discrete-time feedback representation of the system and the application of the well-known engineering tools of harmonic balance to approximate the emerging solutions. More specifically, the results are a higher-order extension of the previous developments obtained by the authors for nondegenerate bifurcations. Two examples are included for illustration.

Data Processing in Bifurcation Analysis of Maps with Time-Delays in the Frequency Domain

2014 ◽  
Vol 685 ◽  
pp. 634-637
Author(s):  
Li Zeng ◽  
Jun Wei Wang
Keyword(s):  
Data Processing ◽  
Frequency Domain ◽  
Time Delays ◽  
Period Doubling ◽  
Feedback Systems ◽  
Bifurcation Solution ◽  
Nonlinear Maps ◽  
The Stability ◽  
Frequency Domain Approach ◽  
Period Doubling Bifurcations

A unified frequency-domain approach to analyze the NS (Neimark-Sacker) bifurcations and the period-doubling bifurcations of nonlinear maps with time-delays in the linear feed-forward term is presented. The technique relies on the HBA (harmonic balance approximation, a very important method in data processing ) and feedback systems theory. The expressions of the bifurcation solution and the stability are derived.

DETECTION OF LIMIT CYCLE BIFURCATIONS USING HARMONIC BALANCE METHODS

2004 ◽  
Vol 14 (10) ◽  
pp. 3647-3654 ◽  
Author(s):  
FEDERICO I. ROBBIO ◽  
DIEGO M. ALONSO ◽  
JORGE L. MOIOLA
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Vector Fields ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Period Doubling ◽  
Nonlinear Feedback ◽  
Feedback Systems ◽  
Zero Eigenvalue ◽  
Complex Singularities

In this paper, bifurcations of limit cycles close to certain singularities of the vector fields are explored using an algorithm based on the harmonic balance method, the theory of nonlinear feedback systems and the monodromy matrix. Period-doubling, pitchfork and Neimark–Sacker bifurcations of cycles are detected close to a Gavrilov–Guckenheimer singularity in two modified Rössler systems. This special singularity has a zero eigenvalue and a pair of pure imaginary eigenvalues in the linearization of the flow around its equilibrium. The presented results suggest that the proposed technique can be promising in analyzing limit cycle bifurcations arising in the unfoldings of other complex singularities.

scholarly journals Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Şeyma Bılazeroğlu ◽  
Huseyin Merdan ◽  
Luca Guerrini
Keyword(s):  
Hopf Bifurcation ◽  
Differential Equations ◽  
Discrete Time ◽  
Time Delays ◽  
Nonlinear Differential Equations ◽  
Dynamical Behavior ◽  
Reaction System ◽  
Hopf Bifurcations ◽  
Normal Form Theory ◽  
Center Manifold Reduction

<p style='text-indent:20px;'>Hopf bifurcations of a Lengyel-Epstein model involving two discrete time delays are investigated. First, stability analysis of the model is given, and then the conditions on parameters at which the system has a Hopf bifurcation are determined. Second, bifurcation analysis is given by taking one of delay parameters as a bifurcation parameter while fixing the other in its stability interval to show the existence of Hopf bifurcations. The normal form theory and the center manifold reduction for functional differential equations have been utilized to determine some properties of the Hopf bifurcation including the direction and stability of bifurcating periodic solution. Finally, numerical simulations are performed to support theoretical results. Analytical results together with numerics present that time delay has a crucial role on the dynamical behavior of Chlorine Dioxide-Iodine-Malonic Acid (CIMA) reaction governed by a system of nonlinear differential equations. Delay causes oscillations in the reaction system. These results are compatible with those observed experimentally.</p>

HOPF BIFURCATIONS AND DEGENERACIES IN CHUA'S CIRCUIT — A PERSPECTIVE FROM A FREQUENCY DOMAIN APPROACH

1999 ◽  
Vol 09 (01) ◽  
pp. 295-303 ◽  
Author(s):  
JORGE L. MOIOLA ◽  
LEON O. CHUA
Keyword(s):  
Frequency Domain ◽  
Harmonic Balance ◽  
Amplitude Equation ◽  
Hopf Bifurcations ◽  
Chua’S Circuit ◽  
Nonlinear Circuit ◽  
Explicit Formulas ◽  
Chua's Circuit ◽  
Lyapunov Index ◽  
Frequency Domain Approach

In the present work explicit formulas for analyzing the birth of limit cycles arising in the Chua's circuit through a Hopf bifurcation is provided. A local amplitude equation is derived using a frequency domain approach and harmonic balance approximations. Furthermore, the first Lyapunov index used to detect degenerate Hopf bifurcations is derived in terms of the parameters of the nonlinear circuit. A perspective for analyzing other bifurcations using this frequency domain approach is discussed.

Aerodynamic Asymmetry Analysis of Unsteady Flows in Turbomachinery

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Kivanc Ekici ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Harmonic Balance ◽  
Coupling Effect ◽  
Unsteady Flows ◽  
Unsteady Aerodynamic ◽  
Blade Row ◽  
Balance Technique ◽  
Turbomachinery Blades ◽  
Aerodynamic Asymmetry ◽  
Frequency Domain Approach ◽  
Harmonic Balance Technique

A nonlinear harmonic balance technique for the analysis of aerodynamic asymmetry of unsteady flows in turbomachinery is presented. The present method uses a mixed time-domain/frequency-domain approach that allows one to compute the unsteady aerodynamic response of turbomachinery blades to self-excited vibrations. Traditionally, researchers have investigated the unsteady response of a blade row with the assumption that all the blades in the row are identical. With this assumption the entire wheel can be modeled using complex periodic boundary conditions and a computational grid spanning a single blade passage. In this study, the steady/unsteady aerodynamic asymmetry is modeled using multiple passages. Specifically, the method has been applied to aerodynamically asymmetric flutter problems for a rotor with a symmetry group of 2. The effect of geometric asymmetries on the unsteady aerodynamic response of a blade row is illustrated. For the cases investigated in this paper, the change in the diagonal terms (blade on itself) dominated the change in stability. Very little mode coupling effect caused by the off-diagonal terms was found.

Behavior of a Self-Sustained Electromechanical Transducer and Routes to Chaos

2005 ◽  
Vol 128 (3) ◽  
pp. 282-293 ◽  
Author(s):  
J. C. Chedjou ◽  
K. Kyamakya ◽  
I. Moussa ◽  
H.-P. Kuchenbecker ◽  
W. Mathis
Keyword(s):  
Electronic Circuit ◽  
Initial Conditions ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Electromechanical System ◽  
Phase Portraits ◽  
Coupling Coefficients ◽  
Oscillatory Solutions ◽  
Electromechanical Transducer ◽  
Design Engineers

This paper studies the dynamics of a self-sustained electromechanical transducer. The stability of fixed points in the linear response is examined. Their local bifurcations are investigated and different types of bifurcation likely to occur are found. Conditions for the occurrence of Hopf bifurcations are derived. Harmonic oscillatory solutions are obtained in both nonresonant and resonant cases. Their stability is analyzed in the resonant case. Various bifurcation diagrams associated to the largest one-dimensional (1-D) numerical Lyapunov exponent are obtained, and it is found that chaos can appear suddenly, through period doubling, period adding, or torus breakdown. The extreme sensitivity of the electromechanical system to both initial conditions and tiny variations of the coupling coefficients is also outlined. The experimental study of̱the electromechanical system is carried out. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the electromechanical system. Correspondences are established between the coefficients of the electromechanical system model and the components of the electronic circuit. Harmonic oscillatory solutions and phase portraits are obtained experimentally. One of the most important contributions of this work is to provide a set of reliable analytical expressions (formulas) describing the electromechanical system behavior. These formulas are of great importance for design engineers as they can be used to predict the states of the electromechanical systems and respectively to avoid their destruction. The reliability of the analytical formulas is demonstrated by the very good agreement with the results obtained by both the numeric and the experimental analysis.

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.

Second Order Averaging Study of an Autoparametric System

1993 ◽  
Author(s):  
Bappaditya Banerjee ◽  
Anil K. Bajaj ◽  
Patricia Davies
Keyword(s):  
Dynamical Behavior ◽  
Period Doubling ◽  
Nonlinear Effects ◽  
Single Mode ◽  
Pitchfork Bifurcation ◽  
Second Order ◽  
System Response ◽  
Response Curves ◽  
Vibratory System ◽  
First Order

Abstract The autoparametric vibratory system consisting of a primary spring-mass-dashpot system coupled with a damped simple pendulum serves as an useful example of two degree-of-freedom nonlinear systems that exhibit complex dynamic behavior. It exhibits 1:2 internal resonance and amplitude modulated chaos under harmonic forcing conditions. First-order averaging studies of this system using AUTO and KAOS have yielded useful information about the amplitude dynamics of this system. Response curves of the system indicate saturation and the pitchfork bifurcation sets are found to be symmetric. The period-doubling route to chaotic solutions is observed. However questions about the range of the small parameter ε (a function of the forcing amplitude) for which the solutions are valid cannot be answered by a first-order study. Some observed dynamical behavior, like saturation, may not persist when higher-order nonlinear effects are taken into account. Second-order averaging of the system, using Mathematica (Maeder, 1991; Wolfram, 1991) is undertaken to address these questions. Loss of saturation is observed in the steady-state amplitude responses. The breaking of symmetry in the various bifurcation sets becomes apparent as a consequence of ε appearing in the averaged equations. The dynamics of the system is found to be very sensitive to damping, with extremely complicated behavior arising for low values of damping. For large ε second-order averaging predicts additional Pitchfork and Hopf bifurcation points in the single-mode response.

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