Linear Scaling Laws in Bifurcations of Scalar Maps

1994 ◽  
Vol 49 (12) ◽  
pp. 1207-1211 ◽  
Author(s):  
Celso Grebogi
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Scaling Laws ◽  
Linear Scaling ◽  
Two Dimensional ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Canonical Map ◽  
Global Scaling

Abstract A global scaling property for bifurcation diagrams of periodic orbits of smooth scalar maps with both one and two dimensional parameter spaces is examined. It is argued that for both parameter spaces bifurcations within a periodic window of a given scalar map are well approximated by a linear transformation of the bifurcation diagram of a canonical map.

COMPLEX BIFURCATION STRUCTURES IN THE HINDMARSH–ROSE NEURON MODEL

2007 ◽  
Vol 17 (09) ◽  
pp. 3071-3083 ◽  
Author(s):  
J. M. GONZÀLEZ-MIRANDA
Keyword(s):  
Phase Diagram ◽  
Bifurcation Diagram ◽  
Parameter Space ◽  
Neuron Model ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Rapid Responses ◽  
Bifurcation Structures

The results of a study of the bifurcation diagram of the Hindmarsh–Rose neuron model in a two-dimensional parameter space are reported. This diagram shows the existence and extent of complex bifurcation structures that might be useful to understand the mechanisms used by the neurons to encode information and give rapid responses to stimulus. Moreover, the information contained in this phase diagram provides a background to develop our understanding of the dynamics of interacting neurons.

The bandcount increment scenario. I. Basic structures

2008 ◽  
Vol 464 (2095) ◽  
pp. 1867-1883 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Basic Structure ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Bifurcation Structures ◽  
Multi Band

Bifurcation structures in two-dimensional parameter spaces formed only by chaotic attractors are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In the first part, we report a novel bifurcation scenario formed by crises bifurcations, which includes multi-band chaotic attractors with arbitrary high bandcounts and determines the basic structure of the chaotic domain.

The bandcount increment scenario. II. Interior structures

2008 ◽  
Vol 464 (2097) ◽  
pp. 2247-2263 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Parameter Space ◽  
Complex Interaction ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in the two-dimensional parameter spaces formed by chaotic attractors alone are still far away from being understood completely. In a series of three papers, we investigate the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In this second part, we investigate fine substructures nested into the basic structures reported and explained in part I. It is demonstrated that the overall structure of the chaotic domain is caused by a complex interaction of bandcount increment, bandcount adding and bandcount doubling structures, whereby some of them are nested into each other ad infinitum leading to self-similar structures in the parameter space.

The bandcount increment scenario. III. Deformed structures

2008 ◽  
Vol 465 (2101) ◽  
pp. 41-57 ◽  
Author(s):  
Viktor Avrutin ◽  
Bernd Eckstein ◽  
Michael Schanz
Keyword(s):  
Canonical Form ◽  
Chaotic Attractors ◽  
Two Dimensional ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
One Dimensional ◽  
Practical Applications ◽  
Chaotic Region ◽  
Self Similar ◽  
Bifurcation Structures

Bifurcation structures in two-dimensional parameter spaces formed by chaotic attractors alone are still a long way from being understood completely. In a series of three papers, we investigated the chaotic domain without periodic inclusions for a map, which is considered by many authors as some kind of one-dimensional canonical form for discontinuous maps. In Part I, the basic structures in the chaotic region are explained by the bandcount increment scenario. In Part II, fine self-similar substructures nested into the bandcount increment scenario are explained by the bandcount-adding and -doubling scenarios, nested into each other ad infinitum. Hereby, we fixed in both previous parts one of the parameters to a non-generic value, and studied the remaining two-dimensional parameter subspace. In this Part III, finally we investigated the structures under variation of this third parameter. Remarkably, this step is the most important with respect to practical applications, since it cannot be expected that these operate exactly at the previously investigated specific value.

Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity

2010 ◽  
Vol 20 (2) ◽  
pp. 023103 ◽  
Author(s):  
Cristiane Stegemann ◽  
Holokx A. Albuquerque ◽  
Paulo C. Rech
Keyword(s):  
Two Dimensional ◽  
Cubic Nonlinearity ◽  
Dimensional Parameter ◽  
Parameter Spaces

Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

2014 ◽  
Vol 24 (07) ◽  
pp. 1430020 ◽  
Author(s):  
Paulo C. Rech
Keyword(s):  
Dynamical Systems ◽  
Continuous Time ◽  
Nonlinear Dynamical Systems ◽  
Angular Frequency ◽  
Two Dimensional ◽  
Periodic Forcing ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Nonlinear Dynamical ◽  
First Order

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

scholarly journals Data-driven discovery of dimensionless numbers and scaling laws from experimental measurements

2021 ◽  
Author(s):  
Xiaoyu Xie ◽  
Wing Kam Liu ◽  
Zhengtao Gan
Keyword(s):  
Scaling Laws ◽  
Synthetic Data ◽  
Experimental Measurements ◽  
Dimensionless Numbers ◽  
Dimensional Parameter ◽  
Parameter Spaces ◽  
Design And Optimization ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard ◽  
Similitude Theory

Abstract Dimensionless numbers and scaling laws provide elegant insights into the characteristic properties of physical systems. Classical dimensional analysis and similitude theory fail to identify a set of unique dimensionless numbers for a highly-multivariable system with incomplete governing equations. In this study, we embed the principle of dimensional invariance into a two-level machine learning scheme to automatically discover dominant and unique dimensionless numbers and scaling laws from data. The proposed methodology, called dimensionless learning, can be treated as a physics-based dimension reduction. It can reduce high-dimensional parameter spaces into descriptions involving just a few physically-interpretable dimensionless parameters, which significantly simplifies the process design and optimization of the system. We demonstrate the algorithm by solving several challenging engineering problems with noisy experimental measurements (not synthetic data) collected from the literature. The examples include turbulent Rayleigh-Bénard convection, vapor depression dynamics in laser melting of metals, and porosity formation in 3D printing. We also show that the proposed approach can identify dimensionally homogeneous differential equations with minimal parameters by leveraging sparsity-promoting techniques.

Bifurcation Diagram of a Map with Multiple Critical Points

2018 ◽  
Vol 28 (05) ◽  
pp. 1850065 ◽  
Author(s):  
M. Romera ◽  
G. Pastor ◽  
M.-F. Danca ◽  
A. Martin ◽  
A. B. Orue ◽  
...  
Keyword(s):  
Bifurcation Diagram ◽  
Periodic Orbits ◽  
Critical Points ◽  
Bifurcation Diagrams

In this work a conjecture to draw the bifurcation diagram of a map with multiple critical points is enunciated. The conjecture is checked by using two quartic maps in order to verify that the bifurcation diagrams obtained according to the conjecture contain all the periodic orbits previously counted by Xie and Hao for maps with four laps. We show that a map with split bifurcation contains more periodic orbits than those counted by these authors for a map with the same number of laps.

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