Bifurcations and Arnold Tongues of a Multiplier-Accelerator Model

2021 ◽  
Author(s):  
Jiyu Zhong
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Flip Bifurcation ◽  
Policy Rule ◽  
Parameter Region ◽  
Invariant Circle ◽  
Arnold Tongue ◽  
Investment Function ◽  
Arnold Tongues ◽  
Stable Invariant

Abstract In this paper, we investigate the bifurcations of a multiplier-acceler-ator model with nonlinear investment function in an anti-cyclical fiscal policy rule. Firstly, we give the conditions that the model produces supercritical flip bifurcation and subcritical one respectively. Secondly, we prove that the model undergoes a generalized flip bifurcation and present a parameter region such that the model possesses two 2-periodic orbits. Thirdly, it is proved that the model undergoes supercritical Neimark-Sacker bifurcation and produces an attracting invariant circle surrounding a fixed point. Fourthly, we present the Arnold tongues such that the model has periodic orbits on the invariant circle produced from the Neimark-Sacker bifurcation. Finally, to verify the correctness of our results, we numerically simulate a attracting 2-periodic orbit, an stable invariant circle, an Arnold tongue with rotation number 1/7 and an attracting 7-periodic orbit on the invariant circle.

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

scholarly journals Time averaged properties along unstable periodic orbits and chaotic orbits in two map systems

2008 ◽  
Vol 15 (4) ◽  
pp. 675-680 ◽  
Author(s):  
Y. Saiki ◽  
M. Yamada
Keyword(s):  
Fluid Dynamics ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Space Physics ◽  
Unstable Periodic Orbits ◽  
Unstable Periodic Orbit ◽  
Geophysical Fluid Dynamics ◽  
Chaotic Orbits ◽  
Low Dimensional ◽  
Complex Phenomena

Abstract. Unstable periodic orbit (UPO) recently has become a keyword in analyzing complex phenomena in geophysical fluid dynamics and space physics. In this paper, sets of UPOs in low dimensional maps are theoretically or systematically found, and time averaged properties along UPOs are studied, in relation to those of chaotic orbits.

DYNAMICS OF THE MUTHUSWAMY–CHUA SYSTEM

2013 ◽  
Vol 23 (08) ◽  
pp. 1350136 ◽  
Author(s):  
YUANFAN ZHANG ◽  
XIANG ZHANG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Electronic Circuit ◽  
Small Perturbations ◽  
Chaotic Phenomena ◽  
Invariant Surfaces

The Muthuswamy–Chua system [Formula: see text] describes the simplest electronic circuit which can have chaotic phenomena. In this paper, we first prove the existence of three families of consecutive periodic orbits of the system when α = 0, two of which are located on consecutive invariant surfaces and form two invariant topological cylinders. Then we prove that for α > 0 if the system has a periodic orbit or a chaotic attractor, it must intersect both of the planes z = 0 and z = -1 infinitely many times as t tends to infinity. As a byproduct, we get an example of unstable invariant topological cylinders which are not normally hyperbolic and which are also destroyed under small perturbations.

Global Bifurcation Approximation in Controlling Chaotic Systems

1998 ◽  
Vol 08 (05) ◽  
pp. 1013-1023
Author(s):  
Byoung-Cheon Lee ◽  
Bong-Gyun Kim ◽  
Bo-Hyeun Wang
Keyword(s):  
Feedback Control ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Chaotic Attractor ◽  
Global Bifurcation ◽  
Good Control ◽  
Hysteresis Phenomenon ◽  
Unstable Periodic Orbits ◽  
Tracking Method ◽  
Adaptive Tracking

In our previous research [Lee et al., 1995], we demonstrated that return map control and adaptive tracking method can be used together to locate, stabilize and track unstable periodic orbits (UPO) automatically. Our adaptive tracking method is based on the control bifurcation (CB) phenomenon which is another route to chaos generated by feedback control. Along the CB route, there are numerous driven periodic orbits (DPOs), and they can be good control targets if small system modification is allowed. In this paper, we introduce a new control concept of global bifurcation approximation (GBA) which is quite different from the traditional local linear approximation (LLA). Based on this approach, we also demonstrate that chaotic attractor can be induced from a periodic orbit. If feedback control is applied along the direction to chaos, small erratic fluctuations of a periodic orbit is magnified and the chaotic attractor is induced. One of the special features of CB is the existence of irreversible orbit (IO) which is generated at the strong extreme of feedback control and has irreversible property. We show that IO induces a hysteresis phenomenon in CB, and we discuss how to keep away from IO.

CODIMENSION 3 HETEROCLINIC BIFURCATIONS WITH ORBIT AND INCLINATION FLIPS IN REVERSIBLE SYSTEMS

2008 ◽  
Vol 18 (12) ◽  
pp. 3689-3701 ◽  
Author(s):  
YANCONG XU ◽  
DEMING ZHU ◽  
FENGJIE GENG
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Homoclinic Orbit ◽  
Heteroclinic Orbit ◽  
Homoclinic Bifurcation ◽  
Reversible Systems ◽  
Reversible System ◽  
Bifurcation Diagrams ◽  
Symmetric Periodic Orbits ◽  
The Continuum

Heteroclinic bifurcations with orbit-flips and inclination-flips are investigated in a four-dimensional reversible system by using the method originally established in [Zhu, 1998; Zhu & Xia, 1998]. The existence and coexistence of R-symmetric homoclinic orbit and R-symmetric heteroclinic orbit, R-symmetric homoclinic orbit and R-symmetric periodic orbit are obtained. The double R-symmetric homoclinic bifurcation is found, and the continuum of R-symmetric periodic orbits accumulating into a homoclinic orbit is also demonstrated. Moreover, the bifurcation surfaces and the existence regions are given, and the corresponding bifurcation diagrams are drawn.

scholarly journals Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point

1985 ◽  
Vol 5 (4) ◽  
pp. 501-517 ◽  
Author(s):  
Lluís Alsedà ◽  
Jaume Llibre ◽  
Michał Misiurewicz ◽  
Carles Simó
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Rotation Interval ◽  
Continuous Map ◽  
Continuous Maps

AbstractLet f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.

Representations of Artin’s braid groups and linking numbers of periodic orbits

1995 ◽  
Vol 04 (02) ◽  
pp. 197-212 ◽  
Author(s):  
John Guaschi
Keyword(s):  
Lower Bound ◽  
Periodic Orbit ◽  
Topological Entropy ◽  
Periodic Orbits ◽  
Braid Groups ◽  
Matrix Representations ◽  
Linking Number ◽  
Linking Numbers ◽  
Small Period

Let P be a periodic orbit of period n≥3 of an orientation-preserving homeomorphism f of the 2-disc. Let q be the least integer greater than or equal to n/2−1. Then f admits a periodic orbit Q of period less than or equal to q such that the linking number of P about Q is non-zero. This answers a question of Franks in the affirmative in the case that P has small period. We also derive a result regarding matrix representations of Artin’s braid groups. Finally a lower bound for the topological entropy of a braid in terms of the trace of its Burau matrix is found.

scholarly journals Tree maps with non divisible periodic orbits

1997 ◽  
Vol 56 (3) ◽  
pp. 467-471 ◽  
Author(s):  
Xiangdong Ye
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Small Perturbation ◽  
Number Of Ends ◽  
Tree Map

Let End (T) be the number of ends of a treeTandf:T→Tbe continuous. We show thatfhas a non divisible periodic orbit if and only if there are somex∈Tandn> 1 with (n, m) = 1 for each 2 ≤m≤ End(T) such thatx∈ (f(x),fn(x)). Consequently the property of a tree map with a non divisible periodic orbit is preserved under small perturbation.

scholarly journals Cascades of Multiheaded Chimera States for Coupled Phase Oscillators

2014 ◽  
Vol 24 (08) ◽  
pp. 1440014 ◽  
Author(s):  
Yuri L. Maistrenko ◽  
Anna Vasylenko ◽  
Oleksandr Sudakov ◽  
Roman Levchenko ◽  
Volodymyr L. Maistrenko
Keyword(s):  
Periodic Orbit ◽  
Coupled Oscillators ◽  
Phase Oscillators ◽  
Invariant Circle ◽  
Saddle Node Bifurcation ◽  
Chimera States ◽  
Chimera State ◽  
Self Organized ◽  
Saddle Node ◽  
Different Types

Chimera state is a recently discovered dynamical phenomenon in arrays of nonlocally coupled oscillators, that displays a self-organized spatial pattern of coexisting coherence and incoherence. We discuss the appearance of the chimera states in networks of phase oscillators with attractive and with repulsive interactions, i.e. when the coupling respectively favors synchronization or works against it. By systematically analyzing the dependence of the spatiotemporal dynamics on the level of coupling attractivity/repulsivity and the range of coupling, we uncover that different types of chimera states exist in wide domains of the parameter space as cascades of the states with increasing number of intervals of irregularity, so-called chimera's heads. We report three scenarios for the chimera birth: (1) via saddle-node bifurcation on a resonant invariant circle, also known as SNIC or SNIPER, (2) via blue-sky catastrophe, when two periodic orbits, stable and saddle, approach each other creating a saddle-node periodic orbit, and (3) via homoclinic transition with complex multistable dynamics including an "eight-like" limit cycle resulting eventually in a chimera state.

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