Non-trivial wandering domains and homoclinic bifurcations

2001 ◽  
Vol 21 (06) ◽  
Author(s):  
EDUARDO COLLI ◽  
EDSON VARGAS
Keyword(s):  
Homoclinic Bifurcations ◽  
Wandering Domains

scholarly journals Minimal Universal Model for Chaos in Laser with Feedback

2021 ◽  
Vol 31 (04) ◽  
pp. 2130013
Author(s):  
Riccardo Meucci ◽  
Stefano Euzzor ◽  
F. Tito Arecchi ◽  
Jean-Marc Ginoux
Keyword(s):  
Laser Physics ◽  
Nonlinear Oscillator ◽  
Three Dimensional ◽  
Universal Model ◽  
Type Formula ◽  
The Third ◽  
Proposed Model ◽  
Homoclinic Bifurcations ◽  
Electronic Implementation ◽  
Different Time Scales

We revisit the model of the laser with feedback and the minimal nonlinearity leading to chaos. Although the model has its origin in laser physics, with peculiarities related to the [Formula: see text] laser, it belongs to the class of the three-dimensional paradigmatic nonlinear oscillator models giving chaos. The proposed model contains three key nonlinearities, two of which are of the type [Formula: see text], where [Formula: see text] and [Formula: see text] are the fast and slow variables. The third one is of the type [Formula: see text], where [Formula: see text] is an intermediate feedback variable. We analytically demonstrate that it is essential for producing chaos via local or global homoclinic bifurcations. Its electronic implementation in the range of kilo Hertz region confirms its potential in describing phenomena evolving on different time scales.

scholarly journals Escaping Fatou components of transcendental self-maps of the punctured plane

2019 ◽  
pp. 1-25 ◽  
Author(s):  
DAVID MARTÍ-PETE
Keyword(s):  
Approximation Theory ◽  
Entire Functions ◽  
Holomorphic Functions ◽  
Sufficient Condition ◽  
Wandering Domain ◽  
Simply Connected ◽  
Fatou Components ◽  
Wandering Domains

Abstract We study the iteration of transcendental self-maps of $\,\mathbb{C}^*\!:=\mathbb{C}\setminus \{0\}$ , that is, holomorphic functions $f:\mathbb{C}^*\to\mathbb{C}^*$ for which both zero and infinity are essential singularities. We use approximation theory to construct functions in this class with escaping Fatou components, both wandering domains and Baker domains, that accumulate to $\{0,\infty\}$ in any possible way under iteration. We also give the first explicit examples of transcendental self-maps of $\,\mathbb{C}^*$ with Baker domains and with wandering domains. In doing so, we developed a sufficient condition for a function to have a simply connected escaping wandering domain. Finally, we remark that our results also provide new examples of entire functions with escaping Fatou components.

scholarly journals Iteration of certain meromorphic functions with unbounded singular values

2009 ◽  
Vol 30 (3) ◽  
pp. 877-891 ◽  
Author(s):  
TARAKANTA NAYAK ◽  
M. GURU PREM PRASAD
Keyword(s):  
Real Line ◽  
Critical Parameter ◽  
Meromorphic Functions ◽  
Period Doubling ◽  
Inverse Function ◽  
Singular Values ◽  
Fatou Set ◽  
Unbounded Subset ◽  
Simply Connected ◽  
Wandering Domains

AbstractLet ℳ={f(z)=(zm/sinhm z) for z∈ℂ∣ either m or m/2 is an odd natural number}. For eachf∈ℳ, the set of singularities of the inverse function offis an unbounded subset of the real line ℝ. In this paper, the iteration of functions in one-parameter family 𝒮={fλ(z)=λf(z)∣λ∈ℝ∖{0}} is investigated for eachf∈ℳ. It is shown that, for eachf∈ℳ, there is a critical parameterλ*>0 depending onfsuch that a period-doubling bifurcation occurs in the dynamics of functionsfλin 𝒮 when the parameter |λ| passes throughλ*. The non-existence of Baker domains and wandering domains in the Fatou set offλis proved. Further, it is shown that the Fatou set offλis infinitely connected for 0<∣λ∣≤λ*whereas for ∣λ∣≥λ*, the Fatou set offλconsists of infinitely many components and each component is simply connected.

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