scholarly journals Design of a New Chaotic System Based on Van Der Pol Oscillator and Its Encryption Application

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 743 ◽  
Author(s):  
Jianbin He ◽  
Jianping Cai
Keyword(s):  
Control Method ◽  
Variable Parameter ◽  
Stable Limit Cycle ◽  
Simple Algorithm ◽  
Van Der Pol Oscillator ◽  
Stable Limit ◽  
Van Der Pol ◽  
Dynamical Characteristics ◽  
New Chaotic System ◽  
Exclusive Or

The Van der Pol oscillator is investigated by the parameter control method. This method only needs to control one parameter of the Van der Pol oscillator by a simple periodic function; then, the Van der Pol oscillator can behave chaotically from the stable limit cycle. Based on the new Van der Pol oscillator with variable parameter (VdPVP), some dynamical characteristics are discussed by numerical simulations, such as the Lyapunov exponents and bifurcation diagrams. The numerical results show that there exists a positive Lyapunov exponent in the VdPVP. Therefore, an encryption algorithm is designed by the pseudo-random sequences generated from the VdPVP. This simple algorithm consists of chaos scrambling and chaos XOR (exclusive-or) operation, and the statistical analyses show that it has good security and encryption effectiveness. Finally, the feasibility and validity are verified by simulation experiments of image encryption.

Bonhoeffer-van der Pol Oscillator Model of the Sino-Atrial Node: A Possible Mechanism of Heart Rate Regulation

1994 ◽  
Vol 33 (01) ◽  
pp. 116-119 ◽  
Author(s):  
S. Sato ◽  
S. Doi ◽  
T. Nomura
Keyword(s):  
Heart Rate ◽  
Stable Limit Cycle ◽  
Nonlinear Dynamic System ◽  
Van Der Pol Oscillator ◽  
Transition Curve ◽  
Stable Limit ◽  
Van Der Pol ◽  
Rate Regulation ◽  
Dynamic System Theory ◽  
Heart Rate Regulation

Abstract:A Bonhoeffer-van der Pol equation with a stable limit cycle is proposed as a model of the pacemaker in the sino-atrial node to exptain heart rate regulation. Standard tools, such as the phase transition curve in nonlinear dynamic system theory, are used to analyze the model and results are compared with other studies on experiments with dogs.

Developing a Reduced-Order Model to Understand Non-Synchronous Vibration (NSV) in Turbomachinery

2012 ◽  
Author(s):  
Stephen T. Clark ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Stable Limit Cycle ◽  
Forced Response ◽  
Van Der Pol Oscillator ◽  
Future Research ◽  
Limit Cycle Oscillation ◽  
Preliminary Design ◽  
Stable Limit ◽  
Van Der Pol ◽  
Reduced Order ◽  
Van Der Pol Model

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model Non-Synchronous Vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. The results also show the maximum amplitude of the LCO occurs at an off-resonant condition, i.e., when the natural shedding frequency of the aerodynamic instability is not coincident with the natural modal frequency of the structure. This conclusion is especially relevant in preliminary design in industry because it suggests that design engineers cannot treat NSV as a normal Campbell-diagram crossing as they would for preliminary design for forced response; it is possible that by redesigning the blade, the response amplitude of the blade may actually be higher. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper Orthogonal Decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. Both the modeling of experimental NSV and preliminary design development will occur in future research.

Extending a Van der Pol Based Reduced-Order Model for Fluid-Structure Interaction Applied to Non-Synchronous Vibrations in Turbomachinery

2021 ◽  
Author(s):  
Richard Hollenbach ◽  
Robert Kielb ◽  
Kenneth Hall
Keyword(s):  
Fluid Structure Interaction ◽  
Reduced Order Model ◽  
Van Der Pol Oscillator ◽  
Previous Model ◽  
Stable Limit ◽  
Order Model ◽  
Van Der Pol ◽  
Fluid Structure ◽  
Reduced Order ◽  
Structure Interaction

Abstract This paper expands upon a multi-degree-of-freedom, Van der Pol oscillator used to model buffet and Nonsynchronous Vibrations (NSV) in turbines. Two degrees-of-freedom are used, a fluid tracking variable incorporating a Van der Pol oscillator and a classic spring, mass, damper mounted cylinder variable; thus, this model is one of fluid-structure interaction. This model has been previously shown to exhibit the two main aspects of NSV. The first is the lock-in or entrainment phenomenon of the fluid shedding frequency jumping onto the natural frequency of the oscillator, while the second is a stable limit cycle oscillation (LCO) once the transient solution disappears. Improvements are made to the previous model to better understand this aeroelastic phenomenon. First, an error minimizing technique through a system identification method is used to tune the coefficients in the Reduced Order Model (ROM) to improve the accuracy in comparison to experimental data. Secondly, a cubic stiffness term is added to the fluid equation; this term is often seen in the Duffing Oscillator equation, which allows this ROM to capture the experimental behavior more accurately, seen in previous literature. The finalized model captures the experimental cylinder data found in literature much better than the previous model. These improvements also open the door for future models, such as that of a pitching airfoil or a turbomachinery blade, to create a preliminary design tool for studying NSV in turbomachinery.

CONTROLLING UNCERTAIN VAN DER POL OSCILLATOR VIA ROBUST NONLINEAR FEEDBACK CONTROL

2004 ◽  
Vol 14 (05) ◽  
pp. 1671-1681 ◽  
Author(s):  
MAO-YIN CHEN ◽  
ZHENG-ZHI HAN ◽  
YUN SHANG ◽  
GUANG-DENG ZONG
Keyword(s):  
Feedback Control ◽  
Control Method ◽  
Reference Signal ◽  
Variable Structure ◽  
Van Der Pol Oscillator ◽  
Nonlinear Feedback Control ◽  
Nonlinear Feedback ◽  
Van Der Pol ◽  
Feedback Control Method ◽  
System Uncertainties

Combining the backstepping design and the variable structure control, we propose a robust nonlinear feedback control method to control an uncertain van der Pol oscillator even if there exist system uncertainties and external disturbances in this oscillator. If system uncertainties are estimated and some parameters are chosen suitably, the output of van der Pol osicllator can track arbitrary smooth reference signal. Theoretical analysis and numerical simulations verify the effectiveness of this method.

scholarly journals STUDY POINTS OF REST HEREDITARITY DYNAMIC SYSTEMS VAN DER POL-DUFFING

2019 ◽  
pp. 71-77
Author(s):  
Е.Р. Новикова ◽  
Р.И. Паровик
Keyword(s):  
Limit Cycle ◽  
Dynamic Systems ◽  
Limit Cycles ◽  
Stable Limit Cycle ◽  
Viscous Friction ◽  
Oscillatory System ◽  
Stable Limit ◽  
Van Der Pol ◽  
Nonlinear Oscillatory System ◽  
Stable Limit Cycles

Using numerical modeling, oscillograms and phase trajectories were constructed to study the limit cycles of a van der Pol Duffing nonlinear oscillatory system with a power memory. The simulation results showed that in the absence of a power memory (α = 2, β = 1) or the classical van der Pol Duffing dynamical system, there is a single stable limit cycle, i.e. Lienar theorem holds. In the case of viscous friction (α = 2, 0 < β < 1), there is a family of stable limit cycles of various shapes. In other cases, the limit cycle is destroyed in two scenarios: a Hopf bifurcation (limit cycle-limit point) or (limit cycle-aperiodic process). Further continuation of the research may be related to the construction of the spectrum of Lyapunov maximal exponents in order to identify chaotic oscillatory regimes for the considered hereditary dynamic system (HDS). В работе с помощью численного моделирования построены осциллограммы и фазовые траектории с целью исследования предельных циклов нелинейной колебательной системы Ван-дер-Поля Дуффинга со степенной памятью. Результаты моделирования показали, что в случае отсутствия степенной памяти (α = 2, β = 1) или классической динамической системы Ван-дер-Поля Дуффинга, существует единственный устойчивый предельный цикл, т.е. выполняется теорема Льенара. В случае вязкого трения (α = 2, 0 < β < 1), существует семейство устойчивых предельных циклов различной формы. В остальных случаях происходит разрушение предельного цикла по двум сценариям: бифуркация Хопфа (предельный цикл-предельная точка) или (предельный циклапериодический процесс). Дальнейшее продолжение исследований может быть связано с построением спектра максимальных показателей Ляпунова с целью идентификации хаотических колебательных режимов для рассматриваемой эредитарной динамической системы (ЭДС).

STABLE OSCILLATIONS AND DEVIL'S STAIRCASE IN THE VAN DER POL OSCILLATOR

2000 ◽  
Vol 10 (01) ◽  
pp. 155-164 ◽  
Author(s):  
T. GILBERT ◽  
R. W. GAMMON
Keyword(s):  
Rotation Number ◽  
Variable Parameter ◽  
Relaxation Oscillator ◽  
Van Der Pol Oscillator ◽  
Driving Frequency ◽  
Devil's Staircase ◽  
Van Der Pol ◽  
Devil’S Staircase ◽  
Staircase Structure ◽  
Parameter Values

A forced van der Pol relaxation oscillator is studied experimentally in the regime of stable oscillations. The variable parameter is chosen to be the driving frequency. For a range of parameter values, we show that the rotation number varies continuously from 0 to 1. This work provides experimental evidence that period-adding bifurcations to chaos previously reported by Kennedy and Chua are intimately connected to the existence of a regime of stable oscillations where the rotation number shows a Devil's-staircase structure.

ON SOME MATHEMATICAL TOPICS IN CLASSICAL SYNCHRONIZATION.: A TUTORIAL

2004 ◽  
Vol 14 (07) ◽  
pp. 2143-2160 ◽  
Author(s):  
ANDREY SHILNIKOV ◽  
LEONID SHILNIKOV ◽  
DMITRY TURAEV
Keyword(s):  
Complex Dynamics ◽  
Invariant Tori ◽  
Global Bifurcation ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Homoclinic Loop ◽  
Van Der Pol ◽  
Mathematical Problems ◽  
Parameter Case ◽  
Synchronization Theory

A few mathematical problems arising in the classical synchronization theory are discussed; especially those relating to complex dynamics. The roots of the theory originate in the pioneering experiments by van der Pol and van der Mark, followed by the theoretical studies by Cartwright and Littlewood. Today, we focus specifically on the problem on a periodically forced stable limit cycle emerging from a homoclinic loop to a saddle point. Its analysis allows us to single out the regions of simple and complex dynamics, as well as to yield a comprehensive description of bifurcational phenomena in the two-parameter case. Of a particular value is the global bifurcation of a saddle-node periodic orbit. For this bifurcation, we prove a number of theorems on birth and breakdown of nonsmooth invariant tori.

COEXISTENCE OF POINT, PERIODIC AND STRANGE ATTRACTORS

2013 ◽  
Vol 23 (05) ◽  
pp. 1350093 ◽  
Author(s):  
JULIEN CLINTON SPROTT ◽  
XIONG WANG ◽  
GUANRONG CHEN
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Strange Attractor ◽  
Chaotic Attractor ◽  
Stable Equilibrium ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
New Chaotic System ◽  
Point Attractor ◽  
New System

For a dynamical system described by a set of autonomous ordinary differential equations, an attractor can be a point, a periodic cycle, or even a strange attractor. Recently, a new chaotic system with only one stable equilibrium was described, which locally converges to the stable equilibrium but is globally chaotic. This paper further shows that for certain parameters, besides the point attractor and chaotic attractor, this system also has a coexisting stable limit cycle, demonstrating that this new system is truly complicated and interesting.

Parameter Estimation of a Fluttering Aeroelastic System in the Transitional Reynolds Number Regime

2010 ◽  
Author(s):  
Mohammad Khalil ◽  
Abhijit Sarkar ◽  
Dominique Poirel
Keyword(s):  
Reynolds Number ◽  
Parameter Estimation ◽  
Wind Tunnel ◽  
Van Der Pol Oscillator ◽  
Limit Cycle Oscillation ◽  
Parameter Estimates ◽  
Stable Limit ◽  
Van Der Pol ◽  
Aeroelastic System ◽  
Rigid Wing

We report the parameter estimation results of a self-sustaining aeroelastic oscillator. The system is composed of a rigid wing that is elastically mounted on a rig, which in turn is fixed in a wind tunnel. For certain flow conditions, in particular dictated by the Reynolds number in the transitional regime, the wing extracts energy from the flow leading to a stable limit cycle oscillation. The basic physical mechanism at the origin of the oscillations is laminar boundary layer separation, which leads to negative aerodynamic damping. An empirical model of the aeroelastic system is proposed in the form of a generalized Duffing-van der Pol oscillator, whereby the linear and nonlinear aeroelastic terms are unknowns to be estimated. The model (input) noise process accounting for the amplitude modulation observed from experiments will also be estimated. We apply a Bayesian inference based batch data assimilation method in tackling this strongly nonlinear and non-Gaussian model. In particular, Markov Chain Monte Carlo sampling technique is used to generate samples from the joint distribution of the unknown parameters given noisy measurement data. The extended Kalman filter is utilized to obtain the conditional distribution of the model state given the noisy measurements. The parameter estimates for a third order generalized Duffing-van der Pol oscillator are obtained and marginal and joint probability density functions for the parameters will be presented for both a numerical model and a rigid wing that is elastically mounted on a rig in a wind tunnel.

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