scholarly journals Complex Dynamics and Synchronization in a System of Magnetically Coupled Colpitts Oscillators

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
L. K. Kana ◽  
A. Fomethe ◽  
H. B. Fotsin ◽  
E. T. Wembe ◽  
A. I. Moukengue
Keyword(s):  
Complex Dynamics ◽  
Spectral Characteristics ◽  
Period Doubling ◽  
Magnetic Coupling ◽  
Coupled System ◽  
Coupling Factor ◽  
Coupled Systems ◽  
Numerical Exploration ◽  
The Stability ◽  
Parameter Ranges

We propose the use of a simple, cheap, and easy technique for the study of dynamic and synchronization of the coupled systems: effects of the magnetic coupling on the dynamics and of synchronization of two Colpitts oscillators (wireless interaction). We derive a smooth mathematical model to describe the dynamic system. The stability of the equilibrium states is investigated. The coupled system exhibits spectral characteristics such as chaos and hyperchaos in some parameter ranges of the coupling. The numerical exploration of the dynamics system reveals various bifurcations scenarios including period-doubling and interior crisis transitions to chaos. Moreover, various interesting dynamical phenomena such as transient chaos, coexistence of solution, and multistability (hysteresis) are observed when the magnetic coupling factor varies. Theoretical reasons for such phenomena are provided and experimentally confirmed with practical measurements in a wireless transfer.

INTERMITTENCY AND CHAOS NEAR HOPF BIFURCATION WITH BROKEN 𝕆(2) × 𝕆(2) SYMMETRY

2013 ◽  
Vol 23 (08) ◽  
pp. 1350139
Author(s):  
YANG ZOU ◽  
GERHARD DANGELMAYR ◽  
IULIANA OPREA
Keyword(s):  
Hopf Bifurcation ◽  
Phase Space ◽  
Normal Form ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Coupled System ◽  
Dimensional Model ◽  
Low Dimensional

The eight-dimensional normal form for a Hopf bifurcation with 𝕆(2) × 𝕆(2) symmetry is perturbed by imperfection terms that break a continuous translation symmetry. The parameters of the fully symmetric normal form are fixed to values for which all basic periodic solutions residing in two-dimensional fixed point subspaces are unstable, and the dynamics is attracted by a chaotic attractor resulting from a period doubling cascade of periodic orbits. By using symmetry-adapted variables, the dimension of the phase space of the normal form is reduced to four and the dimension of the perturbed normal form is reduced to five. In the reduced phase space, periodic solutions are revealed as fixed points, and quasiperiodic solutions as periodic orbits. For the perturbed normal form, parameter regimes with different types of chaotic dynamics are identified when the imperfection parameter is varied. The characteristics of this complex dynamics are symmetry breaking and increasing, various period doubling cascades, intermittency and crises, and switching between symmetry-conjugated chaotic saddles. In particular, the perturbed system serves as a low dimensional model for the complicated switching dynamics found in simulations of the globally coupled system of Ginzburg–Landau equations extending the 𝕆(2) × 𝕆(2)-symmetric normal form to account for spatial modulations. In addition, this system can be considered as a low dimensional model for the dynamics of perturbed waves in anisotropic systems with imperfect geometries due to the presence of sidewalls.

scholarly journals Hopf Bifurcation, Hopf-Hopf Bifurcation, and Period-Doubling Bifurcation in a Four-Species Food Web

2018 ◽  
Vol 2018 ◽  
pp. 1-21
Author(s):  
Huayong Zhang ◽  
Ju Kang ◽  
Tousheng Huang ◽  
Xuebing Cong ◽  
Shengnan Ma ◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Food Web ◽  
Complex Dynamics ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
Food Web Structure ◽  
Center Manifold Theorem ◽  
Dynamical Complexity ◽  
Period 2 ◽  
The Stability

Complex dynamics of a four-species food web with two preys, one middle predator, and one top predator are investigated. Via the method of Jacobian matrix, the stability of coexisting equilibrium for all populations is determined. Based on this equilibrium, three bifurcations, i.e., Hopf bifurcation, Hopf-Hopf bifurcation, and period-doubling bifurcation, are analyzed by center manifold theorem, bifurcation theorem, and numerical simulations. We reveal that, influenced by the three bifurcations, the food web can exhibit very complex dynamical behaviors, including limit cycles, quasiperiodic behaviors, chaotic attractors, route to chaos, period-doubling cascade in orbits of period 2, 4, and 8 and period 3, 6, and 12, periodic windows, intermittent period, and chaos crisis. However, the complex dynamics may disappear with the extinction of one of the four populations, which may also lead to collapse of the food web. It suggests that the dynamical complexity and food web stability are determined by the food web structure and existing populations.

BIFURCATION AND PREDICTABILITY ANALYSIS OF A LOW-ORDER ATMOSPHERIC CIRCULATION MODEL

1995 ◽  
Vol 05 (06) ◽  
pp. 1701-1711 ◽  
Author(s):  
A. SHIL'NIKOV ◽  
G. NICOLIS ◽  
C. NICOLIS
Keyword(s):  
Atmospheric Circulation ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Winter Season ◽  
Period Doubling ◽  
Circulation Model ◽  
Predictability Analysis ◽  
Parameter Ranges ◽  
Parameter Values ◽  
Low Order

A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out. It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation. The principal mechanisms leading to the appearance of complex dynamics around this bifurcation are described and various routes to chaotic behavior are identified, such as the transition through the period doubling cascade, the breakdown of an invariant torus and homoclinic bifurcations of a saddle-focus. Non-trivial limit sets in the form of a chaotic attractor or a chaotic repeller are found in some parameter ranges. Their presence implies an enhanced unpredictability of the system for parameter values corresponding to the winter season.

Institutional Reforms and Interaction of Local Governments

2015 ◽  
Vol 3 (2) ◽  
pp. 25-33
Author(s):  
Georges Sarafopoulos ◽  
Panagiotis G. Ioannidis
Keyword(s):  
Nash Equilibrium ◽  
Local Governments ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Replicator Dynamics ◽  
Discrete Dynamical System ◽  
Period Doubling ◽  
Box Dimension ◽  
Existence And Stability ◽  
The Stability

The paper considers the interaction between regions during the implementation of a reform, on regional development through a discrete dynamical system based on replicator dynamics. The existence and stability of equilibria of this system are studied. The authors show that the parameter of the local prosperity may change the stability of equilibrium and cause a structure to behave chaotically. For the low values of this parameter the game has a stable Nash equilibrium. Increasing these values, the Nash equilibrium becomes unstable, through period-doubling bifurcation. The complex dynamics, bifurcations and chaos are displayed by computing numerically Lyapunov numbers, sensitive dependence on initial conditions and the box dimension.

scholarly journals Complex Dynamics in a Nonlinear Cobweb Model for Real Estate Market

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Junhai Ma ◽  
Lingling Mu
Keyword(s):  
Real Estate ◽  
Demand Function ◽  
Complex Dynamics ◽  
Initial Conditions ◽  
Period Doubling ◽  
Real Estate Market ◽  
Delayed Feedback Control ◽  
Cobweb Model ◽  
Sensitive Dependence ◽  
The Stability

We establish a nonlinear real estate model based on cobweb theory, where the demand function and supply function are quadratic. The stability conditions of the equilibrium are discussed. We demonstrate that as some parameters varied, the stability of Nash equilibrium is lost through period-doubling bifurcation. The chaotic features are justified numerically via computing maximal Lyapunov exponents and sensitive dependence on initial conditions. The delayed feedback control (DFC) method is applied to control the chaos of system.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

scholarly journals A Finite Element Approach to the Analysis of Rotating Bladed-Disk Assemblies Coupled With Flexible Shaft

1994 ◽  
Author(s):  
Wen Zhang ◽  
Wenliang Wang ◽  
Hao Wang ◽  
Jiong Tang
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Coupled System ◽  
Equation Of Motion ◽  
Coupled Systems ◽  
The Finite Element Method ◽  
Bladed Disk ◽  
Modal Synthesis ◽  
Element Approach ◽  
Final Equation

A method for dynamic analysis of flexible bladed-disk/shaft coupled systems is presented in this paper. Being independant substructures first, the rigid-disk/shaft and each of the bladed-disk assemblies are analyzed separately in a centrifugal force field by means of the finite element method. Then through a modal synthesis approach the equation of motion for the integral system is derived. In the vibration analysis of the rotating bladed-disk substructure, the geometrically nonlinear deformation is taken into account and the rotationally periodic symmetry is utilized to condense the degrees of freedom into one sector. The final equation of motion for the coupled system involves the degrees of freedom of the shaft and those of only one sector of each of the bladed-disks, thereby reducing the computer storage. Some computational and experimental results are given.

Turning Dynamics: Part 1 — Experimental Analysis

2012 ◽  
Author(s):  
Eric B. Halfmann ◽  
C. Steve Suh ◽  
N. P. Hung
Keyword(s):  
Instantaneous Frequency ◽  
Period Doubling ◽  
Displacement Sensor ◽  
Laser Displacement Sensor ◽  
Turning Process ◽  
Time Frequency ◽  
Spectral Components ◽  
Tool Vibrations ◽  
Exact Moment ◽  
The Stability

The workpiece and tool vibrations in a lathe are experimentally studied to establish improved understanding of cutting dynamics that would support efforts in exceeding the current limits of the turning process. A Keyence laser displacement sensor is employed to monitor the workpiece and tool vibrations during chatter-free and chatter cutting. A procedure is developed that utilizes instantaneous frequency (IF) to identify the modes related to measurement noise and those innate of the cutting process. Instantaneous frequency is shown to thoroughly characterize the underlying turning dynamics and identify the exact moment in time when chatter fully developed. That IF provides the needed resolution for identifying the onset of chatter suggests that the stability of the process should be monitored in the time-frequency domain to effectively detect and characterize machining instability. It is determined that for the cutting tests performed chatters of the workpiece and tool are associated with the changing of the spectral components and more specifically period-doubling bifurcation. The analysis presented provides a view of the underlying dynamics of the lathe process which has not been experimentally observed before.

MASTER STABILITY FUNCTIONS FOR SYNCHRONIZED COUPLED SYSTEMS

1999 ◽  
Vol 09 (12) ◽  
pp. 2315-2320 ◽  
Author(s):  
LOUIS M. PECORA ◽  
THOMAS L. CARROLL
Keyword(s):  
Simple Form ◽  
Coupled Systems ◽  
Stability Function ◽  
Coupled Oscillator ◽  
Synchronous State ◽  
Master Stability Function ◽  
Stability Functions ◽  
The Stability

We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function which solves, once and for all, the problem of synchronous stability for many couplings of that oscillator.

scholarly journals A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ebenezer Bonyah ◽  
Isaac Dontwi ◽  
Farai Nyabadza
Keyword(s):  
Human Population ◽  
Reproduction Number ◽  
Buruli Ulcer ◽  
Coupled System ◽  
Model Parameters ◽  
The Public ◽  
Treatment Function ◽  
Medical Facilities ◽  
The Stability ◽  
Saturated Treatment

The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources, delays in treatment, and macilent capacity in medical facilities. These challenges limit the number of infected individuals that access medical facilities. While most of the mathematical models with treatment assume a treatment function proportional to the number of infected individuals, in settings with such limitations, this assumption may not be valid. To capture these challenges, a mathematical model of the Buruli ulcer with a saturated treatment function is developed and studied. The model is a coupled system of two submodels for the human population and the environment. We examine the stability of the submodels and carry out numerical simulations. The model analysis is carried out in terms of the reproduction number of the submodel of environmental dynamics. The dynamics of the human population submodel, are found to occur at the steady states of the submodel of environmental dynamics. Sensitivity analysis is carried out on the model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment. The model suggests that more effort should be focused on environmental management. The paper is concluded by discussing the public implications of the results.

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