LOCAL AND GLOBAL BIFURCATIONS IN THREE-DIMENSIONAL, CONTINUOUS, PIECEWISE SMOOTH MAPS

2011 ◽  
Vol 21 (06) ◽  
pp. 1617-1636 ◽  
Author(s):  
SOMA DE ◽  
PARTHA SHARATHI DUTTA ◽  
SOUMITRO BANERJEE ◽  
AKHIL RANJAN ROY
Keyword(s):  
Three Dimensional ◽  
Global Bifurcation ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Numerical Continuation ◽  
Homoclinic Tangency ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Border Collision Bifurcation ◽  
Smooth Map

In this work, we study the dynamics of a three-dimensional, continuous, piecewise smooth map. Much of the nontrivial dynamics of this map occur when its fixed point or periodic orbit hits the switching manifold resulting in the so-called border collision bifurcation. We study the local and global bifurcation phenomena resulting from such borderline collisions. The conditions for the occurrence of nonsmooth period-doubling, saddle-node, and Neimark–Sacker bifurcations are derived. We show that dangerous border collision bifurcation can also occur in this map. Global bifurcations arise in connection with the occurrence of nonsmooth Neimark–Sacker bifurcation by which a spiral attractor turns into a saddle focus. The global dynamics are systematically explored through the computation of resonance tongues and numerical continuation of mode-locked invariant circles. We demonstrate the transition to chaos through the breakdown of mode-locked torus by degenerate period-doubling bifurcation, homoclinic tangency, etc. We show that in this map a mode-locked torus can be transformed into a quasiperiodic torus if there is no global bifurcation.

Bifurcation of Quasiperiodic Orbit in a 3D Piecewise Linear Map

2017 ◽  
Vol 27 (10) ◽  
pp. 1730033 ◽  
Author(s):  
Mahashweta Patra ◽  
Soumitro Banerjee
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Bundle Method ◽  
Piecewise Linear Map ◽  
Border Collision Bifurcations ◽  
Unstable Manifolds ◽  
Smooth Map ◽  
Smooth System

Earlier investigations have demonstrated how a quasiperiodic orbit in a three-dimensional smooth map can bifurcate into a quasiperiodic orbit with two disjoint loops or into a quasiperiodic orbit of double the length in the shape of a Möbius strip. Using a three-dimensional piecewise smooth (PWS) normal form map, we show that in a piecewise smooth system, in addition to the mechanisms reported earlier, new pathways of creation of tori with multiple loops may result from border collision bifurcations. We also illustrate the occurrence of multiple attractor bifurcations due to the interplay between the stable and the unstable manifolds. Two techniques of analyzing bifurcations of ergodic tori are available in literature: the second Poincaré section method and the Lyapunov bundle method. We have shown that these methods can explain the period-doubling and double covering bifurcations in PWS systems, but fail in some cases — especially those which result from nonsmoothness of the system. We have shown that torus bifurcations due to border collision can be explained by change in eigenvalues of the unstable fixed points.

Global Dynamics in the Leslie–Gower Model with the Allee Effect

2018 ◽  
Vol 28 (12) ◽  
pp. 1850151 ◽  
Author(s):  
Valery A. Gaiko ◽  
Cornelis Vuik
Keyword(s):  
Singular Point ◽  
Bifurcation Analysis ◽  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Global Dynamics ◽  
Global Bifurcations ◽  
Biomedical System

We complete the global bifurcation analysis of the Leslie–Gower system with the Allee effect which models the dynamics of the populations of predators and their prey in a given ecological or biomedical system. In particular, studying global bifurcations of limit cycles, we prove that such a system can have at most two limit cycles surrounding one singular point.

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Bifurcation and Chaos in Synchronous Manifold of a Forest Model

2015 ◽  
Vol 25 (12) ◽  
pp. 1550157
Author(s):  
Chun-Ming Huang ◽  
Jonq Juang
Keyword(s):  
Fixed Point ◽  
Positive Integer ◽  
Period Doubling ◽  
Global Dynamics ◽  
Energy Depletion ◽  
Forest Model ◽  
Finite Period ◽  
Smooth Map ◽  
Two Parameters ◽  
Attracting Fixed Point

In previous papers [Isagi et al., 1997; Satake & Iwasa, 2000], a forest model was proposed. The authors demonstrated numerically that the mature forest could possibly exhibit annual reproduction (fixed point synchronization), periodic and chaotic synchronization as the energy depletion constant [Formula: see text] is gradually increased. To understand such rich synchronization phenomena, we are led to study global dynamics of a piecewise smooth map [Formula: see text] containing two parameters [Formula: see text] and [Formula: see text]. Here [Formula: see text] is the energy depletion quantity and [Formula: see text] is the coupling strength. In particular, we obtain the following results. First, we prove that [Formula: see text] has a chaotic dynamic in the sense of Devaney on an invariant set whenever [Formula: see text], which improves a result of [Chang & Chen, 2011]. Second, we prove, via the Schwarzian derivative and a generalized result of [Singer, 1978], that [Formula: see text] exhibits the period adding bifurcation. Specifically, we show that for any [Formula: see text], [Formula: see text] has a unique global attracting fixed point whenever [Formula: see text] ([Formula: see text]) and that for any [Formula: see text], [Formula: see text] has a unique attracting period [Formula: see text] point whenever [Formula: see text] is less than and near any positive integer [Formula: see text]. Furthermore, the corresponding period [Formula: see text] point instantly becomes unstable as [Formula: see text] moves pass the integer [Formula: see text]. Finally, we demonstrate numerically that there are chaotic dynamics whenever [Formula: see text] is in between and away from two consecutive positive integers. We also observe the route to chaos as [Formula: see text] increases from one positive integer to the next through finite period doubling.

BORDER-COLLISION BIFURCATIONS AND CHAOTIC OSCILLATIONS IN A PIECEWISE-SMOOTH DYNAMICAL SYSTEM

2001 ◽  
Vol 11 (12) ◽  
pp. 2977-3001 ◽  
Author(s):  
ZHANYBAI T. ZHUSUBALIYEV ◽  
EVGENIY A. SOUKHOTERIN ◽  
ERIK MOSEKILDE
Keyword(s):  
Three Dimensional ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Voltage Converter ◽  
Piecewise Smooth Systems ◽  
Border Collision Bifurcations ◽  
Solution Type ◽  
Nonautonomous Differential Equations ◽  
The Subject ◽  
Smooth System

Many problems of engineering and applied science result in the consideration of piecewise-smooth dynamical systems. Examples are relay and pulse-width control systems, impact oscillators, power converters, and various electronic circuits with piecewise-smooth characteristics. The subject of investigation in the present paper is the dynamical model of a constant voltage converter which represents a three-dimensional piecewise-smooth system of nonautonomous differential equations. A specific type of phenomena that arise in the dynamics of piecewise-smooth systems are the so-called border-collision bifurcations. The paper contains a detailed analysis of this type of bifurcational transition in the dynamics of the voltage converter, in particular, the merging and subsequent disappearance of cycles of different types, change of solution type, and period-doubling, -tripling, -quadrupling and -quintupling. We show that a denumerable set of unstable cycles can arise together with stable cycles at border-collision bifurcations. The characteristic peculiarities of border-collision bifurcational transitions in piecewise-smooth systems are described and we provide a comparison with some recent results.

Codimension-One and -Two Bifurcations of a Three-Dimensional Discrete Game Model

2021 ◽  
Vol 31 (02) ◽  
pp. 2150023
Author(s):  
Zohreh Eskandari ◽  
Javad Alidousti ◽  
Reza Khoshsiar Ghaziani
Keyword(s):  
Center Manifold ◽  
Continuation Method ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Continuation ◽  
Game Model ◽  
Codimension Two ◽  
Codimension One ◽  
Two Parameters ◽  
Discrete Game

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.

BIFURCATION SCENARIO OF A THREE-DIMENSIONAL VAN DER POL OSCILLATOR

1993 ◽  
Vol 03 (02) ◽  
pp. 399-404 ◽  
Author(s):  
T. SÜNNER ◽  
H. SAUERMANN
Keyword(s):  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Chaotic Behavior ◽  
Three Dimensional ◽  
Global Bifurcation ◽  
Dynamical Behaviour ◽  
Van Der Pol Oscillator ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Dimensional Models

Nonlinear self-excited oscillations are usually investigated for two-dimensional models. We extend the simplest and best known of these models, the van der Pol oscillator, to a three-dimensional one and study its dynamical behaviour by methods of bifurcation analysis. We find cusps and other local codimension 2 bifurcations. A homoclinic (i.e. global) bifurcation plays an important role in the bifurcation diagram. Finally it is demonstrated that chaos sets in. Thus the system belongs to the few three-dimensional autonomous ones modelling physical situations which lead to chaotic behavior.

scholarly journals Chaotic dynamics of a pulse modulated control system for a heating unit

2018 ◽  
Vol 224 ◽  
pp. 02055
Author(s):  
Yuriy A. Gol’tsov ◽  
Alexander S. Kizhuk ◽  
Vasiliy G. Rubanov
Keyword(s):  
Control System ◽  
Differential Equations ◽  
Chaotic Dynamics ◽  
Period Doubling ◽  
Classical Period ◽  
Nonlinear Phenomena ◽  
Pulse Control ◽  
Heating Unit ◽  
Border Collision Bifurcation ◽  
Period Doubling Bifurcations

The dynamic modes and bifurcations in a pulse control system of a heating unit, the condition of which is described through differential equations with discontinuous right–hand sides, have been studied. It has been shown that the system under research can demonstrate a great variety of nonlinear phenomena and bifurcation transitions, such as quasiperiodicity, multistable behaviour, chaotization of oscillations through a classical period–doubling bifurcations cascade and border–collision bifurcation.

The development of asymmetry and period doubling for oscillatory flow in baffled channels

1996 ◽  
Vol 328 ◽  
pp. 19-48 ◽  
Author(s):  
E. P. L. Roberts ◽  
M. R. Mackley
Keyword(s):  
Reynolds Number ◽  
Oscillatory Flow ◽  
Three Dimensional ◽  
Period Doubling ◽  
Progressive Increase ◽  
Newtonian Flow ◽  
Chaotic Flow ◽  
Spatially Periodic ◽  
Dimensional Simulation ◽  
Time Periodic

We report experimental and numerical observations on the way initially symmetric and time-periodic fluid oscillations in baffled channels develop in complexity. Experiments are carried out in a spatially periodic baffled channel with a sinusoidal oscillatory flow. At modest Reynolds number the observed vortex structure is symmetric and time periodic. At higher values the flow progressively becomes three-dimensional, asymmetric and aperiodic. A two-dimensional simulation of incompressible Newtonian flow is able to follow the flow pattern at modest oscillatory Reynolds number. At higher values we report the development of both asymmetry and a period-doubling cascade leading to a chaotic flow regime. A bifurcation diagram is constructed that can describe the progressive increase in complexity of the flow.

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