Low-Index Equilibrium and Multiple Period-Doubling Cascades to Chaos of Atmospheric Flow in Beta-Plane Channel

2016 ◽  
Vol 26 (08) ◽  
pp. 1630020 ◽  
Author(s):  
Zhi-Min Chen
Keyword(s):  
Hopf Bifurcation ◽  
Equilibrium State ◽  
Dynamical Behavior ◽  
Plane Channel ◽  
Period Doubling ◽  
Flow Patterns ◽  
Beta Plane ◽  
Periodic State ◽  
Stable Periodic ◽  
Driving Source

The nonlinear dynamical behavior of an atmospheric circulation in a beta-plane channel is examined on a five-spectral mode model, truncated from the Charney and DeVore quasi-geostrophic equation. Bifurcation and chaos are observed when subjected to a topographic driving disturbance and a thermally driving zonal source. An equilibrium state undergoes supercritical Hopf bifurcation and becomes a stable periodic state with respect to the magnitude of the thermally driving source, whereas the periodic state undergoes a subcritical Hopf bifurcation and transforms into a low-index equilibrium state with respect to the increasing topographic driving disturbance. The stable periodic state further develops into a pair of stable periodic states when increasing the thermally driving source. The first one with the period of 4.3 days exhibits an oscillation of strong and weak zonal flow patterns, whereas the second one with the period of 6.8 days demonstrates a fluctuation amongst weak zonal disturbance flow patterns. Moreover, the two periodic states transform respectively into chaos through separate period-doubling cascades with the further development of the thermally driving source.

DOUBLE HOPF BIFURCATION FOR STUART–LANDAU SYSTEM WITH NONLINEAR DELAY FEEDBACK AND DELAY-DEPENDENT PARAMETERS

2007 ◽  
Vol 10 (04) ◽  
pp. 423-448 ◽  
Author(s):  
SUQI MA ◽  
QISHAO LU ◽  
S. JOHN HOGAN
Keyword(s):  
Hopf Bifurcation ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Resonant Oscillation ◽  
Double Hopf Bifurcation ◽  
Dependent Parameter ◽  
Stable Periodic ◽  
Delay Feedback Control ◽  
Delay Dependent ◽  
Nonlinear Delay

A Stuart–Landau system under delay feedback control with the nonlinear delay-dependent parameter e-pτ is investigated. A geometrical demonstration method combined with theoretical analysis is developed so as to effectively solve the characteristic equation. Multi-stable regions are separated from unstable regions by allocations of Hopf bifurcation curves in (p,τ) plane. Some weak resonant and non-resonant oscillation phenomena induced by double Hopf bifurcation are discovered. The normal form for double Hopf bifurcation is deduced. The local dynamical behavior near double Hopf bifurcation points are also clarified in detail by using the center manifold method. Some states of two coexisting stable periodic solutions are verified, and some torus-broken procedures are also traced.

scholarly journals Double grazing bifurcations of the non-smooth railway wheelset systems

2021 ◽  
Author(s):  
Pengcheng Miao ◽  
Denghui Li ◽  
Shan Yin ◽  
Jianhua Xie ◽  
Celso Grebogi ◽  
...  
Keyword(s):  
Dynamical Behavior ◽  
Autonomous Systems ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Railway Vehicle ◽  
Impact Model ◽  
Railway Systems ◽  
Discontinuity Mapping ◽  
Stable Periodic ◽  
Grazing Bifurcations

Abstract There are numerous non-smooth factors in railway vehicle systems, such as flange impact, dry friction, creep force, and so on. Such non-smooth factors heavily affect the dynamical behavior of the railway systems. In this paper, we investigate and mathematically analyze the double grazing bifurcations of the railway wheelset systems with flange contact. Two types of models of flange impact are considered, one is a rigid impact model and the other is a soft impact model. First, we derive Poincaré maps near the grazing trajectory by the Poincaré-section discontinuity mapping (PDM) approach for the two impact models. Then, we analyze and compare the near grazing dynamics of the two models. It is shown that in the rigid impact model the stable periodic motion of the railway wheelset system translates into a chaotic motion after the gazing bifurcation, while in the soft impact model a pitchfork bifurcation occurs and the system tends to the chaotic state through a period doubling bifurcation. Our results also extend the applicability of the PDM of one constraint surface to that of two constraint surfaces for autonomous systems.

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 Dynamical analysis of a giving up smoking model with time delay

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Zizhen Zhang ◽  
Ruibin Wei ◽  
Wanjun Xia
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Local Stability ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Form Theory

AbstractIn this paper, we are concerned with a delayed smoking model in which the population is divided into five classes. Sufficient conditions guaranteeing the local stability and existence of Hopf bifurcation for the model are established by taking the time delay as a bifurcation parameter and employing the Routh–Hurwitz criteria. Furthermore, direction and stability of the Hopf bifurcation are investigated by applying the center manifold theorem and normal form theory. Finally, computer simulations are implemented to support the analytic results and to analyze the effects of some parameters on the dynamical behavior of the model.

scholarly journals Dynamical Behavior in a Four-Dimensional Neural Network Model with Delay

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Changjin Xu ◽  
Peiluan Li
Keyword(s):  
Neural Network ◽  
Hopf Bifurcation ◽  
Network Model ◽  
Neural Network Model ◽  
Center Manifold ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Form Theory ◽  
Explicit Formulae ◽  
Delay Differential

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

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.

scholarly journals Bifurcation Analysis of an Age Structured HIV Infection Model with Both Virus-to-Cell and Cell-to-Cell Transmissions

2018 ◽  
Vol 28 (09) ◽  
pp. 1850109 ◽  
Author(s):  
Xiangming Zhang ◽  
Zhihua Liu
Keyword(s):  
T Cells ◽  
Hopf Bifurcation ◽  
Hiv Infection ◽  
Bifurcation Analysis ◽  
Dynamical Behavior ◽  
Infection Model ◽  
Positive Equilibrium ◽  
Semilinear Equations ◽  
Age Structured ◽  
Hiv Infection Model

We make a mathematical analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions to understand the dynamical behavior of HIV infection in vivo. In the model, we consider the proliferation of uninfected CD[Formula: see text] T cells by a logistic function and the infected CD[Formula: see text] T cells are assumed to have an infection-age structure. Our main results concern the Hopf bifurcation of the model by using the theory of integrated semigroup and the Hopf bifurcation theory for semilinear equations with nondense domain. Bifurcation analysis indicates that there exist some parameter values such that this HIV infection model has a nontrivial periodic solution which bifurcates from the positive equilibrium. The numerical simulations are also carried out.

Hopf Bifurcation Induced by Delay Effect in a Diffusive Tumor-Immune System

2018 ◽  
Vol 28 (11) ◽  
pp. 1850136 ◽  
Author(s):  
Ben Niu ◽  
Yuxiao Guo ◽  
Yanfei Du
Keyword(s):  
Immune System ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Center Manifold ◽  
Immune Cell ◽  
Tumor Treatment ◽  
Delay Effect ◽  
Stable Periodic ◽  
The Stability ◽  
Bifurcation And Stability

Tumor-immune interaction plays an important role in the tumor treatment. We analyze the stability of steady states in a diffusive tumor-immune model with response and proliferation delay [Formula: see text] of immune system where the immune cell has a probability [Formula: see text] in killing tumor cells. We find increasing time delay [Formula: see text] destabilizes the positive steady state and induces Hopf bifurcations. The criticality of Hopf bifurcation is investigated by deriving normal forms on the center manifold, then the direction of bifurcation and stability of bifurcating periodic solutions are determined. Using a group of parameters to simulate the system, stable periodic solutions are found near the Hopf bifurcation. The effect of killing probability [Formula: see text] on Hopf bifurcation values is also discussed.

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