Secondary Bifurcations and High Periodic Orbits in Voltage Controlled Buck Converter

1997 ◽  
Vol 07 (12) ◽  
pp. 2755-2771 ◽  
Author(s):  
M. Di Bernardo ◽  
E. Fossas ◽  
G. Olivar ◽  
F. Vasca
Keyword(s):  
Chaotic Attractor ◽  
Invariant Manifolds ◽  
Main Sequence ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Bifurcation Diagrams ◽  
Synchronous Switching ◽  
Route To Chaos ◽  
Secondary Bifurcations

Period doubling route to chaos has been shown to occur in the voltage controlled DC/DC buck converter, both experimentally and numerically. A chaotic attractor was found at the end of the sequence, suddenly followed by an increase of its size. In this paper new secondary bifurcations and high periodic phenomena, coexisting with the main sequence are detected and analyzed over the same range of parameters. A(synchronous)-switching and stroboscopic maps, unstable orbits, bifurcation diagrams, invariant manifolds and basins of attraction are outlined. These tools are put together to reveal the dynamical richness of this nonsmooth system.

scholarly journals Sequence of Routes to Chaos in a Lorenz-Type System

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Fangyan Yang ◽  
Yongming Cao ◽  
Lijuan Chen ◽  
Qingdu Li
Keyword(s):  
Limit Cycles ◽  
Chaotic Attractor ◽  
Type System ◽  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Basins Of Attraction ◽  
Chaotic Attractors ◽  
Main Route ◽  
Route To Chaos ◽  
Period Doubling Bifurcations

This paper reports a new bifurcation pattern observed in a Lorenz-type system. The pattern is composed of a main bifurcation route to chaos (n=1) and a sequence of sub-bifurcation routes with n=3,4,5,…,14 isolated sub-branches to chaos. When n is odd, the n isolated sub-branches are from a period-n limit cycle, followed by twin period-n limit cycles via a pitchfork bifurcation, twin chaotic attractors via period-doubling bifurcations, and a symmetric chaotic attractor via boundary crisis. When n is even, the n isolated sub-branches are from twin period-n/2 limit cycles, which become twin chaotic attractors via period-doubling bifurcations. The paper also shows that the main route and the sub-routes can coexist peacefully by studying basins of attraction.

INTERMITTENCY AND INTERIOR CRISIS AS ROUTE TO CHAOS IN DYNAMIC WALKING OF TWO BIPED ROBOTS

2012 ◽  
Vol 22 (03) ◽  
pp. 1250056 ◽  
Author(s):  
HASSENE GRITLI ◽  
SAFYA BELGHITH ◽  
NAHLA KHRAEIF
Keyword(s):  
Periodic Orbit ◽  
Chaotic Attractor ◽  
Period Doubling ◽  
Biped Robot ◽  
Type I ◽  
Dynamic Walking ◽  
Unstable Periodic Orbit ◽  
Bifurcation Diagrams ◽  
Biped Robots ◽  
Route To Chaos

Recently, passive and semi-passive dynamic walking has been noticed in researches of biped walking robots. Such biped robots are well-known that they demonstrate only a period-doubling route to chaos while walking down sloped surfaces. In previous researches, such route was shown with respect to a continuous change in some parameter of the biped robot. In this paper, two biped robots are introduced: the passive compass-gait biped robot and the semi-passive torso-driven biped robot. The period-doubling scenario route to chaos is revisited for the first biped as the ground slope changes. Furthermore, we will show through bifurcation diagram that the torso-driven biped exhibits also such route to chaos when the slope angle is varied. For such biped, a modified semi-passive control law is introduced in order to stabilize the torso at some desired position. In this work, we will show through bifurcation diagrams that the dynamic walking of the two biped robots reveals two other routes to chaos namely the intermittency route and the interior crisis route. We will stress that the intermittency is generated via a saddle-node bifurcation where an unstable periodic orbit is created. We will highlight that such event takes place for a Type-I intermittency. However, we will emphasize that the interior crisis event occurs when a collision of the unstable periodic orbit with a weak chaotic attractor happens giving rise to a strong chaotic attractor. In addition, we will explore the intermittent step series induced by the interior crisis and also by the Type-I intermittency. In this study, our analysis on chaos and the routes to chaos will be based, beside bifurcation diagrams, on Lyapunov exponents and fractal (Lyapunov) dimension. These two tools are plotted in the parameter space to classify attractors observed in bifurcation diagrams.

CHARACTERIZATION OF CHAOTIC ATTRACTORS INSIDE BAND-MERGING SCENARIO IN A ZAD-CONTROLLED BUCK CONVERTER

2012 ◽  
Vol 22 (10) ◽  
pp. 1230034
Author(s):  
JOHN ALEXANDER TABORDA ◽  
FABIOLA ANGULO ◽  
GERARD OLIVAR
Keyword(s):  
Control Strategy ◽  
Control Parameter ◽  
Period Doubling ◽  
Basins Of Attraction ◽  
Buck Converter ◽  
Chaotic Attractors ◽  
Complex Dynamic ◽  
Self Similar ◽  
Dynamic Links

Zero Average Dynamics (ZAD) control strategy has been developed, applied and widely analyzed in the last decade. Numerous and interesting phenomena have been studied in systems controlled by ZAD strategy. In particular, the ZAD-controlled buck converter has been a source of nonlinear and nonsmooth phenomena, such as period-doubling, merging bands, period-doubling bands, torus destruction, fractal basins of attraction or codimension-2 bifurcations. In this paper, we report a new bifurcation scenario found inside band-merging scenario of ZAD-controlled buck converter. We use a novel qualitative framework named Dynamic Linkcounter (DLC) approach to characterize chaotic attractors between consecutive crisis bifurcations. This approach complements the results that can be obtained with Bandcounter approaches. Self-similar substructures denoted as Complex Dynamic Links (CDLs) are distinguished in multiband chaotic attractors. Geometrical changes in multiband chaotic attractors are detected when the control parameter of ZAD strategy is varied between two consecutive crisis bifurcations. Linkcount subtracting staircases are defined inside band-merging scenario.

Nonlinear Dynamics of Duffing System With Fractional Order Damping

2009 ◽  
Author(s):  
Junyi Cao ◽  
Chengbin Ma ◽  
Hang Xie ◽  
Zhuangde Jiang
Keyword(s):  
Nonlinear Dynamics ◽  
Fractional Order ◽  
Period Doubling ◽  
Excitation Amplitude ◽  
Bifurcation Diagrams ◽  
Duffing System ◽  
Runge Kutta Method ◽  
Route To Chaos ◽  
Euler Methods ◽  
Period Motion

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The four order Runge-Kutta method and ten order CFE-Euler methods are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on the system dynamics is investigated using phase diagrams, bifurcation diagrams and Poincare map. The bifurcation diagram is also used to exam the effects of excitation amplitude and frequency on Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits period motion, chaos, period motion, chaos, period motion in turn when the fractional order changes from 0.1 to 2.0. A period doubling route to chaos is clearly observed.

SIZE RATIO UNIVERSALITY IN CHAOTIC SYSTEMS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250005
Author(s):  
R. D. GALA ◽  
P. PARMANANDA
Keyword(s):  
Nonlinear Systems ◽  
Chaotic Attractor ◽  
Chaotic Systems ◽  
Period Doubling ◽  
Size Ratio ◽  
Route To Chaos

In this paper, we attempt to predict the size of the chaotic attractor at the parameter value where the period tends to infinity for nonlinear systems that follow the period doubling route to chaos, and for which the Feigenbaum universalities hold.

Chaos in Robot Control Equations

1997 ◽  
Vol 07 (03) ◽  
pp. 707-720 ◽  
Author(s):  
Shrinivas Lankalapalli ◽  
Ashitava Ghosal
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
System Of Nonlinear Equations ◽  
Nonlinear Ordinary Differential Equations ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Repetitive Motion ◽  
Route To Chaos

The motion of a feedback controlled robot can be described by a set of nonlinear ordinary differential equations. In this paper, we examine the system of two second-order, nonlinear ordinary differential equations which model a simple two-degree-of-freedom planar robot, undergoing repetitive motion in a plane in the absence of gravity, and under two well-known robot controllers, namely a proportional and derivative controller and a model-based controller. We show that these differential equations exhibit chaotic behavior for certain ranges of the proportional and derivative gains of the controller and for certain values of a parameter which quantifies the mismatch between the model and the actual robot. The system of nonlinear equations are non-autonomous and the phase space is four-dimensional. Hence, it is difficult to obtain significant analytical results. In this paper, we use the Lyapunov exponent to test for chaos and present numerically obtained chaos maps giving ranges of gains and mismatch parameters which result in chaotic motions. We also present plots of the chaotic attractor and bifurcation diagrams for certain values of the gains and mismatch parameters. From the bifurcation diagrams, it appears that the route to chaos is through period doubling.

Sensitivity of routes to chaos in optically injected semiconductor lasers

2014 ◽  
Vol 23 (03) ◽  
pp. 1450036
Author(s):  
Najm M. Al-Hosiny
Keyword(s):  
Semiconductor Laser ◽  
Semiconductor Lasers ◽  
Period Doubling ◽  
Optical Signal ◽  
Optical Injection ◽  
Bifurcation Diagrams ◽  
Routes To Chaos ◽  
Route To Chaos

Two common routes to chaos, period-doubling and quasi-periodic, are theoretically investigated in semiconductor laser subject to optical injection. In particular, the sensitivity of the route to the injection of an additional optical signal is examined using bifurcation diagrams. Period-doubling route to chaos is found to be less sensitive to the perturbation of the second signal than the quasi-periodic route.

scholarly journals Wrinkling instabilities in soft bilayered systems

2017 ◽  
Vol 375 (2093) ◽  
pp. 20160163 ◽  
Author(s):  
Silvia Budday ◽  
Sebastian Andres ◽  
Bastian Walter ◽  
Paul Steinmann ◽  
Ellen Kuhl
Keyword(s):  
Period Doubling ◽  
Media Theory ◽  
Analytical Models ◽  
Control Parameters ◽  
Computational Simulations ◽  
Smart Surfaces ◽  
Post Buckling ◽  
Surface Patterns ◽  
Secondary Instabilities ◽  
Secondary Bifurcations

Wrinkling phenomena control the surface morphology of many technical and biological systems. While primary wrinkling has been extensively studied, experimentally, analytically and computationally, higher-order instabilities remain insufficiently understood, especially in systems with stiffness contrasts well below 100. Here, we use the model system of an elastomeric bilayer to experimentally characterize primary and secondary wrinkling at moderate stiffness contrasts. We systematically vary the film thickness and substrate prestretch to explore which parameters modulate the emergence of secondary instabilities, including period-doubling, period-tripling and wrinkle-to-fold transitions. Our experiments suggest that period-doubling is the favourable secondary instability mode and that period-tripling can emerge under disturbed boundary conditions. High substrate prestretch can suppress period-doubling and primary wrinkles immediately transform into folds. We combine analytical models with computational simulations to predict the onset of primary wrinkling, the post-buckling behaviour, secondary bifurcations and the wrinkle-to-fold transition. Understanding the mechanisms of pattern selection and identifying the critical control parameters of wrinkling will allow us to fabricate smart surfaces with tunable properties and to control undesired surface patterns like in the asthmatic airway. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

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.

NEW PHENOMENA IN THE NEIGHBORHOOD OF THE CODIMENSION-TWO BIFURCATION IN THE TWIN-WELL DUFFING OSCILLATOR

2000 ◽  
Vol 10 (06) ◽  
pp. 1367-1381 ◽  
Author(s):  
W. SZEMPLIŃSKA-STUPNICKA ◽  
A. ZUBRZYCKI ◽  
E. TYRKIEL
Keyword(s):  
Bifurcation Point ◽  
Chaotic Attractor ◽  
Duffing Oscillator ◽  
Codimension Two ◽  
Periodic Attractors ◽  
The Cross ◽  
Complex Phenomena ◽  
New Phenomena ◽  
Secondary Bifurcations ◽  
Codimension Two Bifurcation

In this paper, we study effects of the secondary bifurcations in the neighborhood of the primary codimension-two bifurcation point. The twin-well potential Duffing oscillator is considered and the investigations are focused on the new scenario of destruction of the cross-well chaotic attractor. The phenomenon belongs to the category of the subduction scenario and relies on the replacement of the cross-well chaotic attractor by a pair of unsymmetric 2T-periodic attractors. The exploration of a sequence of accompanying bifurcations throws more light on the complex phenomena that may occur in the neighborhood of the primary codimension-two bifurcation point. It shows that in the close vicinity of the point there appears a transition zone in the system parameter plane, the zone which separates the two so-far investigated scenarios of annihilation of the cross-well chaotic attractor.

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