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.

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.

STABILITY AND HOPF BIFURCATION ON A TWO-NEURON SYSTEM WITH TIME DELAY IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (04) ◽  
pp. 1355-1366 ◽  
Author(s):  
WENWU YU ◽  
JINDE CAO
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Frequency Domain ◽  
Harmonic Balance ◽  
Neuron System ◽  
Bifurcation Theorem ◽  
Nyquist Criterion ◽  
The Stability ◽  
Frequency Domain Approach ◽  
System With Time Delay

In this paper, a general two-neuron model with time delay is considered, where the time delay is regarded as a parameter. It is found that Hopf bifurcation occurs when this delay passes through a sequence of critical value. By analyzing the characteristic equation and using the frequency domain approach, the existence of Hopf bifurcation is determined. The stability of bifurcating periodic solutions are determined by the harmonic balance approach, Nyquist criterion and the graphic Hopf bifurcation theorem. Numerical results are given to justify the theoretical analysis.

CHARACTERIZATION OF STATIC BIFURCATIONS FOR MAPS IN THE FREQUENCY DOMAIN

2007 ◽  
Vol 17 (03) ◽  
pp. 975-983 ◽  
Author(s):  
LI ZENG ◽  
YI ZHAO
Keyword(s):  
Frequency Domain ◽  
Building Blocks ◽  
Complex Singularities ◽  
Smooth Maps ◽  
Continuous Time Systems ◽  
Nonlinear Maps ◽  
Frequency Domain Approach ◽  
Time Systems ◽  
Bifurcation Behavior

In this paper n-dimensional discrete-time systems with static bifurcations are considered from the viewpoint of control theory. This paper presents an adaption of available formulas for bifurcation analysis in two-dimensional continuous-time systems to the case of smooth maps using a frequency domain approach. The analyzed bifurcations are the building blocks to understand other more complex singularities and to propose certain methods for controlling the bifurcation behavior in nonlinear maps in the future.

Double Loop Autopilot Design Based on Frequency Domain

2011 ◽  
Vol 301-303 ◽  
pp. 1724-1729
Author(s):  
Mei Sa Pang ◽  
Deng Hua Li ◽  
Jun Fang Fan ◽  
Xue Fei Li
Keyword(s):  
System Design ◽  
Frequency Domain ◽  
Static Stability ◽  
Double Loop ◽  
Guidance And Control ◽  
Control Gain ◽  
Aerodynamic Parameters ◽  
The Stability ◽  
And Control ◽  
Frequency Domain Approach

The static stability of missile pitching movement is one of the important performances in guidance and control systems. In this paper, a method which consists of the single and double loop longitudinal autopilot using frequency domain approach is proposed to solve the problem efficiently. Single-loop autopilot is used to simplify the system design when the missile is highly static stable; the double-loop autopilot is employed to stabilize the system and improve frequency performance when the missile is static stable or static unstable. Control gain of the system is determined by aerodynamic parameters and frequency domain indexes. The simulation result shows that double-loop autopilot based on frequency domain simplified the system design and improved the stability and robustness of missile system.

scholarly journals Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

2009 ◽  
Vol 4 (3) ◽  
Author(s):  
Eric A. Butcher ◽  
Oleg A. Bobrenkov ◽  
Ed Bueler ◽  
Praveen Nindujarla
Keyword(s):  
Cutting Force ◽  
Extreme Points ◽  
Period Doubling ◽  
Milling Process ◽  
Depth Of Cut ◽  
Approximation Technique ◽  
The Stability ◽  
Delay Differential ◽  
High Degree ◽  
Period Doubling Bifurcations

In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems such as milling are modeled by delay-differential equations with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up- and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found, and an in-depth investigation of the optimal stable immersion levels for down-milling in the vicinity of where the average cutting force changes sign is presented.

Milling Model With Variable Time Delay

2004 ◽  
Author(s):  
X.-H. Long ◽  
B. Balachandran
Keyword(s):  
Time Delay ◽  
Process Model ◽  
Period Doubling ◽  
Milling Process ◽  
Variable Time Delay ◽  
Variable Time ◽  
Stability Of Periodic Orbits ◽  
Loss Of Contact ◽  
The Stability ◽  
Period Doubling Bifurcations

Taking into account the effect of feed rate, a milling-process model with a variable time delay is presented in this article. Loss-of-contact effects are also considered. The development of this formulation is described and the efforts undertaken to examine the stability of periodic orbits of this system are discussed. A semi-discretization treatment is used for the stability analysis, and this analysis provides evidence for period-doubling bifurcations and secondary Hopf bifurcations. Good agreement is found between the numerical results obtained from this work and experimental results published in the literature.

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