Universality of the topology of period doubling dynamical systems

In this work a one-dimensional piecewise-smooth dynamical system, representing a Poincaré return map for dynamical systems of the Lorenz type, is investigated. The system shows a bifurcation scenario similar to the classical period-doubling one, but which is influenced by so-called border collision phenomena and denoted as border collision period-doubling bifurcation scenario. This scenario is formed by a sequence of pairs of bifurcations, whereby each pair consists of a border collision bifurcation and a pitchfork bifurcation. The mechanism leading to this scenario and its characteristic properties, like symmetry-breaking and symmetry-recovering as well as emergence of coexisting attractors, are investigated.

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Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

European Journal of Applied Mathematics ◽

10.1017/s0956792516000115 ◽

2016 ◽

Vol 27 (6) ◽

pp. 904-922 ◽

Cited By ~ 12

Author(s):

STEPHEN COOMBES ◽

RÜDIGER THUL

Keyword(s):

Dynamical Systems ◽

Variational Equation ◽

Period Doubling ◽

Star Graph ◽

Stability Function ◽

Bifurcation Of Limit Cycles ◽

Master Stability Function ◽

Coupled Networks ◽

Low Dimensional ◽

Insight Into

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

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THE ROLE OF CODIMENSION IN DYNAMICAL SYSTEMS

International Journal of Modern Physics C ◽

10.1142/s0129183192000890 ◽

1992 ◽

Vol 03 (06) ◽

pp. 1295-1321 ◽

Cited By ~ 8

Author(s):

JASON A.C. GALLAS

Keyword(s):

Dynamical System ◽

Dynamical Systems ◽

Period Doubling ◽

Main Body ◽

Hénon Map ◽

Henon Map ◽

Dynamical Parameters

Isoperiodic diagrams are used to investigate the topology of the codimension space of a representative dynamical system: the Hénon map. The codimension space is reported to be organized in a simple and regular way: instead of “structures-within-structures” it consists of a “structures-parallel-to-structures” sequence of shrimp-shaped isoperiodic islands immersed on a via caotica. The isoperiodic islands consist of a main body of principal periodicity k=1, 2, 3, 4, …, which bifurcates according to a period-doubling route. The Pk=k×2n, n=0, 1, 2, … shrimps are very densely concentrated along a main α-direction, a neighborhood parallel to the line b=−0.583a+1.025, where a and b are the dynamical parameters in Eq. (1). Isoperiodic diagrams allow to interpret and unify some apparently uncorrelated phenomena, such as ‘period-bubbling’, classes of reverse bifurcations and antimonotonicity and to recognize that they are in fact signatures of the complicated way in which period-doubling occurs in higher codimensional systems.

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NOISE-INDUCED BACKWARD BIFURCATIONS OF STOCHASTIC 3D-CYCLES

Fluctuation and Noise Letters ◽

10.1142/s0219477510000095 ◽

2010 ◽

Vol 09 (01) ◽

pp. 89-106 ◽

Cited By ~ 19

Author(s):

I. BASHKIRTSEVA ◽

L. RYASHKO ◽

P. STIKHIN

Keyword(s):

Dynamical Systems ◽

Limit Cycles ◽

Noise Intensity ◽

Nonlinear Dynamical Systems ◽

Period Doubling ◽

Sensitivity Function ◽

Function Technique ◽

Stochastic Bifurcation ◽

Nonlinear Dynamical ◽

General Method

We study stochastically forced multiple limit cycles of nonlinear dynamical systems in a period-doubling bifurcation zone. Noise-induced transitions between separate parts of the cycle are considered. A phenomenon of a decreasing of the stochastic cycle multiplicity with a noise intensity growth is investigated. We call it by a backward stochastic bifurcation (BSB). In this paper, for the BSB analysis we suggest a stochastic sensitivity function technique. As a result, a method for the estimation of critical values of noise intensity corresponding to BSB is proposed. The constructive possibilities of this general method for the detailed BSB analysis of the multiple stochastic cycles of the forced Roessler system are demonstrated.

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On Instability Leading to Chaos in Dynamical Systems

International Astronomical Union Colloquium ◽

10.1017/s0252921100065921 ◽

1993 ◽

Vol 132 ◽

pp. 47-60

Author(s):

L.M. Saha

Keyword(s):

Dynamical Systems ◽

Strange Attractor ◽

Period Doubling ◽

Stability Conditions ◽

Stability Criteria ◽

Linear Parameter ◽

Standard Map ◽

Taylor Map ◽

Range Of Values ◽

Period Doubling Bifurcations

AbstractInstability of orbits in dynamical systems leading to chaos has been reviewed briefly. Stability criteria for some unimodal mapping which provide various periodic regimes during the period doubling bifurcations has been discussed in detail. Stability conditions are reviewed for standard map (or Chirikov-Taylor map), and results obtained for range of values of the non-linear parameter appearing in the map have been studied. Strange attractor has also been discussed.

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Chaos in Robert Hooke's inverted cone

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2007.1813 ◽

2007 ◽

Vol 463 (2081) ◽

pp. 1259-1269 ◽

Cited By ~ 3

Author(s):

Médéric Argentina ◽

Pierre Coullet ◽

Jean-Marc Gilli ◽

Marc Monticelli ◽

Germain Rousseaux

Keyword(s):

Dynamical Systems ◽

Numerical Simulations ◽

Gravity Field ◽

Chaos Theory ◽

Mechanical Model ◽

Period Doubling ◽

The Sun ◽

Chaotic Motions ◽

Robert Hooke ◽

Classical Experiment

Robert Hooke is perhaps one of the first scientists to have met chaotic motions. Indeed, to invert a cone and let a ball move in it was a mechanical model used by him to mimic the motion of a planet around a centre of force like the Sun. However, as the cone is inclined with respect to the gravity field, the perfect rosace followed by the particle becomes chaotic meanderings. We revisit this classical experiment designed by Hooke with the modern tools of dynamical systems and chaos theory. By a combination of both numerical simulations and experiments, we prove that the scenario of transition to the chaotic behaviour is through a period-doubling instability.

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IMPLEMENTATION OF A CHUA CIRCUIT TO DEMONSTRATE BIFURCATIONS AND STRANGE ATTRACTORS IN A CLASS

Proceedings of the Canadian Engineering Education Association (CEEA) ◽

10.24908/pceea.v0i0.4706 ◽

2012 ◽

Author(s):

Ahmad Byagowi ◽

Witold Kinsner

Keyword(s):

Dynamical Systems ◽

Strange Attractor ◽

Period Doubling ◽

Fractal Dimensions ◽

Strange Attractors ◽

Current Phase ◽

Graduate Course ◽

Chua Circuit ◽

Rényi Dimension ◽

Commercial Off The Shelf

This paper describes the design and implementation of a Chua double-scroll circuit to demonstrate chaos in dynamical systems to students in a graduate course in order to enhance their visualization and understanding of strange attractor and Feigenbaum bifurcation trees.Teaching dynamical systems (i.e., nonlinear systems that can exhibit chaos) is often considered difficult because of the mathematical modeling involved and the inclusion of the fourth strange-attractor state, in addition to the traditional point stability, cyclic stability, and toroidal stability, as found in dynamic systems. A graduate course has been offered at the University of Manitoba for many years to provide both (i) a unified theory of fractal dimensions, together with many practical implementations of algorithms to compute the fractal dimensions, including the Rényi dimension spectrum that is required for characterization of the strange attractors using multifractal analysis.Leon Chua developed a simple nonlinear circuit capable of producing a rich collection of dynamic phenomena, ranging from fixed points to cycle points, standard bifurcations (period doubling), other standard routes to chaos, and chaos itself. The reason for selecting this specific circuit as a class demonstration tool is threefold: (i) the circuit has an analytical model and can be simulated, (ii) the circuit is implementable using available commercial off-the-shelf components, and (iii) the signals in the circuit can be acquired without affecting and altering its operation significantly.This paper describes the architecture, implementation, verification, and testing of the Chua system, as well as an analysis of the data obtained during the current phase of the development. Although there are many possible implementations of Chua’s circuit, our implementation has several innovative design features to make it more applicable to enhance students’ learning in the classroom.

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DYNAMICAL SYSTEMS OF DIFFERENT CLASSES AS MODELS OF THE KICKED NONLINEAR OSCILLATOR

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127401002547 ◽

2001 ◽

Vol 11 (04) ◽

pp. 1065-1077 ◽

Cited By ~ 9

Author(s):

A. P. KUZNETSOV ◽

L. V. TURUKINA ◽

E. MOSEKILDE

Keyword(s):

Dynamical Systems ◽

Parameter Space ◽

Nonlinear Oscillator ◽

Period Doubling ◽

Bifurcation Diagrams ◽

Scaling Properties ◽

Critical Curves

Using the nonlinear dissipative kicked oscillator as an example, the correspondence between the descriptions provided by model dynamical systems of different classes is discussed. A detailed study of the approximate 1D map is undertaken: the period doubling is examined and the possibility of non-Feigenbaum period doubling is shown. Illustrations in the form of bifurcation diagrams and sets of iteration diagrams are given, the scaling properties are demonstrated, and the tricritical points (the terminal points of the Feigenbaum critical curves) in parameter space are found. The congruity with the properties of the corresponding 2D map, the Ikeda map, is studied. A description in terms of tricritical dynamics is found to be adequate only in particular areas of parameter space.

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