First return maps for the dynamics of synaptically coupled conditional bursters

2010 ◽  
Vol 103 (2) ◽  
pp. 87-104 ◽  
Author(s):  
Evandro Manica ◽  
Georgi S. Medvedev ◽  
Jonathan E. Rubin
Keyword(s):  
Return Maps ◽  
First Return Maps

scholarly journals Realization of analytic moduli for parabolic Dulac germs

2021 ◽  
pp. 1-55
Author(s):  
PAVAO MARDEŠIĆ ◽  
MAJA RESMAN
Keyword(s):  
Return Maps ◽  
Analytic Classification ◽  
First Return Maps ◽  
Russian Math

Abstract In a previous paper [P. Mardešić and M. Resman. Analytic moduli for parabolic Dulac germs. Russian Math. Surveys, to appear, 2021, arXiv:1910.06129v2.] we determined analytic invariants, that is, moduli of analytic classification, for parabolic generalized Dulac germs. This class contains parabolic Dulac (almost regular) germs, which appear as first-return maps of hyperbolic polycycles. Here we solve the problem of realization of these moduli.

scholarly journals Symbolic Encoding of Periodic Orbits and Chaos in the Rucklidge System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Chengwei Dong ◽  
Lian Jia ◽  
Qi Jie ◽  
Hantao Li
Keyword(s):  
Periodic Orbits ◽  
Nonlinear Dynamical Systems ◽  
Unstable Periodic Orbits ◽  
Nonlinear Dynamical ◽  
System A ◽  
Return Maps ◽  
Phase Portrait Analysis ◽  
Encoding Method ◽  
First Return Maps ◽  
Parameter Values

To describe and analyze the unstable periodic orbits of the Rucklidge system, a so-called symbolic encoding method is introduced, which has been proven to be an efficient tool to explore the topological properties concealed in these periodic orbits. In this work, the unstable periodic orbits up to a certain topological length in the Rucklidge system are systematically investigated via a proposed variational method. The dynamics in the Rucklidge system are explored by using phase portrait analysis, Lyapunov exponents, and Poincaré first return maps. Symbolic encodings of the periodic orbits with two and four letters based on the trajectory topology in the phase space are implemented under two sets of parameter values. Meanwhile, the bifurcations of the periodic orbits are explored, significantly improving the understanding of the dynamics of the Rucklidge system. The multiple-letter symbolic encoding method could also be applicable to other nonlinear dynamical systems.

THE EFFECT OF SMOOTHING ON THE DYNAMICAL BEHAVIOR OF NOISY CHAOTIC SIGNALS

1993 ◽  
Vol 03 (06) ◽  
pp. 1591-1600
Author(s):  
M. F. H. TARROJA ◽  
V. A. SICAM
Keyword(s):  
Phase Space ◽  
Signal Detection ◽  
Additive Noise ◽  
Dynamical Behavior ◽  
Data Sets ◽  
Gaussian Filter ◽  
Return Maps ◽  
Chaotic Signals ◽  
First Return Maps ◽  
Distinguishing Features

A smoothing, which has the effect of a Gaussian filter, has been performed on noisy chaotic digitized laser signals. It is shown that this smoothing can unravel distinguishing features of chaos in the phase space plots and first return maps and hence is useful in unambiguously differentiating chaos from noise. The calculation of the correlation dimensions reveal that the dynamics of the signals are not drastically affected by the smoothing. The type of smoothing discussed in this paper is useful in analyzing chaotic data sets corrupted by additive noise from the signal detection process.

Bifurcation Structures of Nested Mixed-Mode Oscillations

2021 ◽  
Vol 31 (08) ◽  
pp. 2150121
Author(s):  
Munehisa Sekikawa ◽  
Naohiko Inaba
Keyword(s):  
Mixed Mode ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Mixed Mode Oscillations ◽  
Return Maps ◽  
Mode Oscillation ◽  
Mixed Mode Oscillation ◽  
Bifurcation Structures ◽  
First Return Maps ◽  
Strong Order

In recently published work [Inaba & Kousaka, 2020a; Inaba & Tsubone, 2020b], we discovered significant mixed-mode oscillation (MMO) bifurcation structures in which MMOs are nested. Simple mixed-mode oscillation-incrementing bifurcations (MMOIBs) are known to generate [Formula: see text] oscillations for successive [Formula: see text] between regions of [Formula: see text]- and [Formula: see text]-oscillations, where [Formula: see text] and [Formula: see text] are adjacent simple MMOs, e.g. [Formula: see text] and [Formula: see text], where [Formula: see text] is an integer. MMOIBs are universal phenomena of evidently strong order and have been studied extensively in chemistry, physics, and engineering. Nested MMOIBs are phenomena that are more complex, but have an even stronger order, generating chaotic MMO windows that include sequences [Formula: see text] for successive [Formula: see text], where [Formula: see text] and [Formula: see text] are adjacent MMOIB-generated MMOs, i.e. [Formula: see text] and [Formula: see text] for integer [Formula: see text]. Herein, we investigate the bifurcation structures of nested MMOIB-generated MMOs exhibited by a classical forced Bonhoeffer–van der Pol oscillator. We use numerical methods to prepare two- and one-parameter bifurcation diagrams of the system with [Formula: see text], and 3 for successive [Formula: see text] for the case [Formula: see text]. Our analysis suggests that nested MMOs could be widely observed and are clearly ordered phenomena. We then define the first return maps for nested MMOs, which elucidate the appearance of successively nested MMOIBs.

scholarly journals Genus-one Birkhoff sections for geodesic flows

2014 ◽  
Vol 35 (6) ◽  
pp. 1795-1813 ◽  
Author(s):  
PIERRE DEHORNOY
Keyword(s):  
Tangent Bundle ◽  
Geodesic Flow ◽  
Singular Points ◽  
Geodesic Flows ◽  
Return Maps ◽  
Unit Tangent Bundle ◽  
First Return Maps ◽  
Genus One

We prove that the geodesic flow on the unit tangent bundle to every hyperbolic 2-orbifold that is a sphere with three or four singular points admits explicit genus-one Birkhoff sections, and we determine the associated first return maps.

scholarly journals FURTHER INVESTIGATION OF HYSTERESIS IN CHUA'S CIRCUIT

2002 ◽  
Vol 12 (01) ◽  
pp. 129-134 ◽  
Author(s):  
J. BORRESEN ◽  
S. LYNCH
Keyword(s):  
Numerical Methods ◽  
Feedback Mechanism ◽  
Bistable Behavior ◽  
Nonlinear Resistor ◽  
Chua Circuit ◽  
Return Maps ◽  
Bistable Region ◽  
Physical Medium ◽  
First Return Maps ◽  
Parameter Values

For a system to display bistable behavior (or hysteresis), it is well known that there needs to be a nonlinear component and a feedback mechanism. In the Chua circuit, nonlinearity is supplied by the Chua diode (nonlinear resistor) and in the physical medium, feedback would be inherently present, however, with standard computer models this feedback is omitted. Using Poincaré first return maps, bifurcations for a varying parameter in the Chua circuit equations are investigated for both increasing and decreasing parameter values. Evidence for the existence of a small bistable region is shown and numerical methods are applied to determine the behavior of the solutions within this bistable region.

scholarly journals Improving the Complexity of the Lorenz Dynamics

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-16 ◽  
Author(s):  
María Pilar Mareca ◽  
Borja Bordel
Keyword(s):  
Cross Sections ◽  
Secure Communications ◽  
The Novel ◽  
Return Maps ◽  
Transient Solutions ◽  
The Lorenz System ◽  
The Stability ◽  
First Return Maps ◽  
Chaotic Transient ◽  
New System

A new four-dimensional, hyperchaotic dynamic system, based on Lorenz dynamics, is presented. Besides, the most representative dynamics which may be found in this new system are located in the phase space and are analyzed here. The new system is especially designed to improve the complexity of Lorenz dynamics, which, despite being a paradigm to understand the chaotic dissipative flows, is a very simple example and shows great vulnerability when used in secure communications. Here, we demonstrate the vulnerability of the Lorenz system in a general way. The proposed 4D system increases the complexity of the Lorenz dynamics. The trajectories of the novel system include structures going from chaos to hyperchaos and chaotic-transient solutions. The symmetry and the stability of the proposed system are also studied. First return maps, Poincaré sections, and bifurcation diagrams allow characterizing the global system behavior and locating some coexisting structures. Numerical results about the first return maps, Poincaré cross sections, Lyapunov spectrum, and Kaplan-Yorke dimension demonstrate the complexity of the proposed equations.

Sign in / Sign up

Export Citation Format

Share Document