scholarly journals Bifurcation Analysis of a Van der Pol-Duffing Circuit with Parallel Resistor

2009 ◽  
Vol 2009 ◽  
pp. 1-26 ◽  
Author(s):  
Denis de Carvalho Braga ◽  
Luis Fernando Mello ◽  
Marcelo Messias
Keyword(s):  
Analytical Study ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Van Der Pol ◽  
Codimension One ◽  
Local Bifurcations ◽  
Parameter Values ◽  
Small Periodic

We study the local codimension one, two, and three bifurcations which occur in a recently proposed Van der Pol-Duffing circuit (ADVP) with parallel resistor, which is an extension of the classical Chua's circuit with cubic nonlinearity. The ADVP system presents a very rich dynamical behavior, ranging from stable equilibrium points to periodic and even chaotic oscillations. Aiming to contribute to the understand of the complex dynamics of this new system we present an analytical study of its local bifurcations and give the corresponding bifurcation diagrams. A complete description of the regions in the parameter space for which multiple small periodic solutions arise through the Hopf bifurcations at the equilibria is given. Then, by studying the continuation of such periodic orbits, we numerically find a sequence of period doubling and symmetric homoclinic bifurcations which leads to the creation of strange attractors, for a given set of the parameter values.

scholarly journals MATHEMATICAL MODEL OF THREE SPECIES FOOD CHAIN INTERACTION WITH MIXED FUNCTIONAL RESPONSE

2012 ◽  
Vol 09 ◽  
pp. 334-340 ◽  
Author(s):  
MADA SANJAYA WS ◽  
ISMAIL BIN MOHD ◽  
MUSTAFA MAMAT ◽  
ZABIDIN SALLEH
Keyword(s):  
Mathematical Model ◽  
Functional Response ◽  
Food Chain ◽  
Equilibrium Points ◽  
Dynamical Behavior ◽  
Bifurcation Points ◽  
Dynamical Behaviors ◽  
Practical Life ◽  
Parameter Values ◽  
Chain Interaction

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

DYNAMICS AND MODULATED CHAOS FOR TWO COUPLED OSCILLATORS

2004 ◽  
Vol 14 (01) ◽  
pp. 337-346 ◽  
Author(s):  
QINSHENG BI
Keyword(s):  
Coupled Oscillators ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Period Doubling Bifurcation ◽  
The Other ◽  
Van Der Pol ◽  
Parametrically Excited ◽  
Van Der Pol Oscillators ◽  
Period Doubling Bifurcations

The dynamical behavior of two coupled parametrically excited van der Pol oscillators is investigated in this paper. A special road to chaos is explored in detail. Period-doubling bifurcation associated with one of the frequencies of the system may be observed, the other frequency of the coupled oscillators plays a role in the evolution. It is found that one of the frequencies of the system contributes to the cascade of period-doubling bifurcations associated with the other frequency, which leads to a generalized modulated chaos.

Complex Dynamics in a Harmonically Excited Lennard-Jones Oscillator: Microcantilever-Sample Interaction in Scanning Probe Microscopes1

1998 ◽  
Vol 122 (1) ◽  
pp. 240-245 ◽  
Author(s):  
M. Basso ◽  
L. Giarre´ ◽  
M. Dahleh ◽  
I. Mezic´
Keyword(s):  
Parameter Space ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Operating Conditions ◽  
Invariant Sets ◽  
Jones Potential ◽  
Lennard Jones ◽  
Atomic Force

In this paper we model the microcantilever-sample interaction in an atomic force microscope (AFM) via a Lennard-Jones potential and consider the dynamical behavior of a harmonically forced system. Using nonlinear analysis techniques on attracting limit sets, we numerically verify the presence of chaotic invariant sets. The chaotic behavior appears to be generated via a cascade of period doubling, whose occurrence has been studied as a function of the system parameters. As expected, the chaotic attractors are obtained for values of parameters predicted by Melnikov theory. Moreover, the numerical analysis can be fruitfully employed to analyze the region of the parameter space where no theoretical information on the presence of a chaotic invariant set is available. In addition to explaining the experimentally observed chaotic behavior, this analysis can be useful in finding a controller that stabilizes the system on a nonchaotic trajectory. The analysis can also be used to change the AFM operating conditions to a region of the parameter space where regular motion is ensured. [S0022-0434(00)01401-5]

Hyperchaos and Coexisting Attractors in a Modified van der Pol–Duffing Oscillator

2019 ◽  
Vol 29 (05) ◽  
pp. 1950067 ◽  
Author(s):  
Karthikeyan Rajagopal ◽  
Abdul Jalil M. Khalaf ◽  
Zhouchao Wei ◽  
Viet-Thanh Pham ◽  
Ahmed Alsaedi ◽  
...  
Keyword(s):  
Discrete Model ◽  
Equilibrium Points ◽  
Duffing Oscillator ◽  
Dynamical Behavior ◽  
Coexisting Attractors ◽  
Van Der Pol ◽  
Euler’S Method ◽  
Gate Arrays ◽  
Field Programmable ◽  
Programmable Gate Arrays

This paper deals with a new modified hyperchaotic van der Pol–Duffing (MVPD) snap oscillator. Various dynamical properties of the proposed system are investigated with the help of Lyapunov exponents, stability analysis of the equilibrium points and bifurcation plots. The existence of the Hopf bifurcation is established by analyzing the corresponding characteristic equation. It is also proved that the MVPD oscillator shows multistability with coexisting attractors. Various numerical simulations are conducted and presented to show the dynamical behavior of the MVPD system. To show that the system is hardware realizable, we derive the discrete model of the MVPD system using the Euler’s method and using the hardware–software cosimulation, the proposed MVPD system is implemented in Field Programmable Gate Arrays. It is shown that the output of the digital implementations of the MVPD systems matches the numerical analysis.

MULTIPARAMETRIC BIFURCATIONS IN AN ENZYME-CATALYZED REACTION MODEL

2005 ◽  
Vol 15 (03) ◽  
pp. 905-947 ◽  
Author(s):  
E. FREIRE ◽  
L. PIZARRO ◽  
A. J. RODRÍGUEZ-LUIS ◽  
F. FERNÁNDEZ-SÁNCHEZ
Keyword(s):  
Periodic Orbits ◽  
Dynamical Behavior ◽  
Homoclinic Orbits ◽  
Reaction Model ◽  
Numerical Continuation ◽  
Global Bifurcations ◽  
Codimension One ◽  
Local Bifurcations ◽  
Homoclinic Connections ◽  
Local And Global Bifurcations

An exhaustive analysis of local and global bifurcations in an enzyme-catalyzed reaction model is carried out. The model, given by a planar five-parameter system of autonomous ordinary differential equations, presents a great richness of bifurcations. This enzyme-catalyzed model has been considered previously by several authors, but they only detected a minimal part of the dynamical and bifurcation behavior exhibited by the system. First, we study local bifurcations of equilibria up to codimension-three (saddle-node, cusps, nondegenerate and degenerate Hopf bifurcations, and nondegenerate and degenerate Bogdanov–Takens bifurcations) by using analytical and numerical techniques. The numerical continuation of curves of global bifurcations allows to improve the results provided by the study of local bifurcations of equilibria and to detect new homoclinic connections of codimension-three. Our analysis shows that such a system exhibits up to sixteen different kinds of homoclinic orbits and thirty different configurations of equilibria and periodic orbits. The coexistence of up to five periodic orbits is also pointed out. Several bifurcation sets are sketched in order to show the dynamical behavior the system exhibits. The different codimension-one and -two bifurcations are organized around five codimension-three degeneracies.

scholarly journals Complex dynamics of a three species food-chain model with Holling type IV functional response

2011 ◽  
Vol 16 (3) ◽  
pp. 553-374
Author(s):  
Ranjit Kumar Upadhyay ◽  
Sharada Nandan Raw
Keyword(s):  
Food Chain ◽  
Complex Dynamics ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
Chain Model ◽  
Narrow Region ◽  
Type Iv ◽  
Numerical Bifurcation ◽  
Parameter Values ◽  
Food Chain Model

In this paper, dynamical complexities of a three species food chain model with Holling type IV predator response is investigated analytically as well as numerically. The local and global stability analysis is carried out. The persistence criterion of the food chain model is obtained. Numerical bifurcation analysis reveals the chaotic behavior in a narrow region of the bifurcation parameter space for biologically realistic parameter values of the model system. Transition to chaotic behavior is established via period-doubling bifurcation and some sequences of distinctive period-halving bifurcation leading to limit cycles are observed.

On the Fundamental Issues in Nonstationary Dynamics and Chaos

1993 ◽  
Author(s):  
Ross M. Evan-Iwanowski ◽  
Chu Ho Lu ◽  
Germain L. Ostiguy
Keyword(s):  
Duffing Oscillator ◽  
Dynamical Behavior ◽  
Logistic Map ◽  
Period Doubling ◽  
Codimension One ◽  
Extended Value ◽  
Limit Motion ◽  
Short Time ◽  
Nonstationary Dynamics ◽  
Considerable Complexity

Abstract In nonstationary (NS) systems some control parameters (CPs) have the following forms: CP(t) = CP0 + ψ(t), where CP0 = const, and ψ(t) are arbitrary functions of t. Other arbitrary functions which play a pivotal role in the NS systems are the parameter functions ϕ(CP) = 0, CP = {CP1, CP2...CPn}. While the functions ψ(t) determine the time directions of the NS dynamical behavior, the functions ϕ(CP) = 0 determine the paths for the CPs to follow. The NS processes are permanently transient due to the functions ψ(t) and/or ϕ(CP) = 0, and for that reason, they can be extremely complex. Clearly then, it is essential to address the fundamental problems of cohesion and definitness of these processes. Using select examples, these issues have been studied in this presentation and have been resolved in positive. Specifically, the following have been demonstrated: (1) convergence (definitivness) of the NS logistic map and the softening Duffing oscillator to an NS limit motion (2) the appearance of a sequence of similar attractors for different NS bifurcations (3) the effects of different parameter paths, ϕ(CP) = 0, in the period doubling region of the Duffing oscillator (4) the effects of linear and cyclic paths in transition through the Ueda bifurcation regions. The results obtained show considerable complexity of the NS dynamic and chaotic responses (5) for exponential ψ(t), the Lorenz “weather” three-term model exhibit a periodic “window” in the chaotic range for an extended value of t. (6) the effects of different ψ(t) in the typical codimension one bifurcations (7) the ST chaos may be created or annihilated by injection of NS inputs (8) an efficient and fast stabilization, i.e., reduction of ST vibration to near the static equilibrium in a short time, can be accomplished by NS changes of the parameters of the system.

On the modeling of some triodes-based nonlinear oscillators with complex dynamics: case of the Van der Pol oscillator

2021 ◽  
Author(s):  
Joakim Vianney Ngamsa Tegnitsap ◽  
Merlin Brice Saatsa Tsefack ◽  
Elie Bertrand Megam Ngouonkadi ◽  
Hilaire Bertrand Fotsin
Keyword(s):  
Mathematical Model ◽  
Lyapunov Exponent ◽  
Dynamical Behavior ◽  
Basin Of Attraction ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Oscillator ◽  
Phase Portraits ◽  
Van Der Pol ◽  
Physical Consideration

Abstract In this work, the dynamic of the triode-based Van der Pol oscillator coupled to a linear circuit is investigated (Triode-based VDPCL oscillator). Towards this end, we present a mathematical model of the triode, chosen from among the many different ones present in literature. The dynamical behavior of the system is investigated using classical tools such as two-parameter Lyapunov exponent, one-parameter bifurcation diagram associated with the graph of largest Lyapunov exponent, phase portraits, and time series. Numerical simulations reveal rather rich and complex phenomena including chaos, transient chaos, the coexistence of solutions, crisis, period-doubling followed by reverse period-doubling sequences (bubbles), and bursting oscillation. The coexistence of attractors is illustrated by the phase portraits and the cross-section of the basin of attraction. Such triode-based nonlinear oscillators can find their applications in many areas where ultra-high frequencies and high powers are demanded (radio, radar detection, satellites communication, etc.) since triode can work with these performances and are often used in the aforementioned areas. In contrast to some recent work on triode-based oscillators, LTSPICE simulations, based on real physical consideration of the triode, are carried out in order to validate the theoretical results obtained in this paper as well as the mathematical model adopted for the triode.

BIFURCATION AND CHAOS IN COUPLED BVP OSCILLATORS

2004 ◽  
Vol 14 (04) ◽  
pp. 1305-1324 ◽  
Author(s):  
TETSUSHI UETA ◽  
HISAYO MIYAZAKI ◽  
TAKUJI KOUSAKA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Autonomous System ◽  
Period Doubling ◽  
Chaotic Attractors ◽  
Nonlinear Phenomena ◽  
Van Der Pol ◽  
Bifurcation And Chaos ◽  
Experimental Laboratory ◽  
Chaotic Parameter ◽  
Parameter Values ◽  
The Mathematical Model

Bonhöffer–van der Pol(BVP) oscillator is a classic model exhibiting typical nonlinear phenomena in the planar autonomous system. This paper gives an analysis of equilibria, periodic solutions, strange attractors of two BVP oscillators coupled by a resister. When an oscillator is fixed its parameter values in nonoscillatory region and the others in oscillatory region, create the double scroll attractor due to the coupling. Bifurcation diagrams are obtained numerically from the mathematical model and chaotic parameter regions are clarified. We also confirm the existence of period-doubling cascades and chaotic attractors in the experimental laboratory.

A New Time-Delay Model for Chaotic Glucose-Insulin Regulatory System

2020 ◽  
Vol 30 (12) ◽  
pp. 2050178
Author(s):  
Abdul-Basset A. AL-Hussein ◽  
Fadhil Rahma ◽  
Luigi Fortuna ◽  
Maide Bucolo ◽  
Mattia Frasca ◽  
...  
Keyword(s):  
Time Delay ◽  
Complex Dynamics ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Regulatory System ◽  
Periodic Oscillation ◽  
Insulin Level ◽  
Insulin Degradation ◽  
Delay Model ◽  
New Time

Mathematical modeling is very helpful for noninvasive investigation of glucose-insulin interaction. In this paper, a new time-delay mathematical model is proposed for glucose-insulin endocrine metabolic regulatory feedback system incorporating the [Formula: see text]-cell dynamic and function for regulating and maintaining bloodstream insulin level. The model includes the insulin degradation due to glucose interaction. The dynamical behavior of the model is analyzed and two-dimensional bifurcation diagrams with respect to two essential parameters of the model are obtained. The results show that the time-delay in insulin secretion in response to blood glucose level, and the delay in glucose drop due to increased insulin concentration, can give rise to complex dynamics, such as periodic oscillation. These dynamics are consistent with the biological findings and period doubling cascade and chaotic state which represent metabolic disorder that may lead to diabetes mellitus.

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