On the number of limit cycles in perturbations of a two dimensional nonlinear differential system

Continually expanding periodically translated kernels on the two dimensional grid can yield interesting, beau- tiful and even familiar patterns. For example, expand- ing circular pillbox shaped kernels on a hexagonal grid, adding when there is overlap, yields patterns includ- ing maximally packed circles and a triquetra-type three petal structure used to represent the trinity in Chris- tianity. Continued expansion yields the flower-of-life used extensively in art and architecture. Additional expansion yields an even more interesting emerging ef- florescence of periodic functions. Example images are given for the case of circular pillbox and circular cone shaped kernels. Using Fourier analysis, fundamental properties of these patterns are analyzed. As a func- tion of expansion, some effloresced functions asymp- totically approach fixed points or limit cycles. Most interesting is the case where the efflorescence never repeats. Video links are provided for viewing efflores- cence in real time.

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The number of limit cycles of certain polynomial differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s030821050001341x ◽

1984 ◽

Vol 98 (3-4) ◽

pp. 215-239 ◽

Cited By ~ 66

Author(s):

T. R. Blows ◽

N. G. Lloyd

Keyword(s):

Differential Equations ◽

Limit Cycles ◽

Small Amplitude ◽

Two Dimensional ◽

Differential Systems ◽

Quadratic Systems ◽

Polynomial Differential Equations ◽

Image Position ◽

Cubic Systems

SynopsisTwo-dimensional differential systemsare considered, where P and Q are polynomials. The question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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LIMIT CYCLES OF A DOUBLE OSCILLATOR EXCITED BY DRY FRICTION

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127411030349 ◽

2011 ◽

Vol 21 (10) ◽

pp. 3043-3046 ◽

Cited By ~ 1

Author(s):

SERGEY STEPANOV

Keyword(s):

Fixed Points ◽

Numerical Method ◽

Limit Cycles ◽

Dry Friction ◽

Coulomb Friction ◽

Three Dimensional ◽

Slip Mode ◽

Two Dimensional ◽

Parametric Investigation ◽

Friction Laws

A two-mass oscillator with one mass lying on the driving belt with dry Coulomb friction is considered. A numerical method for finding all limit cycles and their parametric investigation, based on the analysis of fixed points of a two-dimensional map, is suggested. As successive points for the map we chose points of friction transferred from stick mode to slip mode. These transfers are defined by two equalities and yield a two-dimensional map, in contrast to three-dimensional maps that we can construct for regularized continuous dry friction laws.

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A Modified Multistable Chaotic Oscillator

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127418500852 ◽

2018 ◽

Vol 28 (07) ◽

pp. 1850085 ◽

Cited By ~ 35

Author(s):

Zhouchao Wei ◽

Viet-Thanh Pham ◽

Abdul Jalil M. Khalaf ◽

Jacques Kengne ◽

Sajad Jafari

Keyword(s):

Chaotic System ◽

Limit Cycles ◽

Initial Conditions ◽

Chaotic Oscillator ◽

Two Dimensional ◽

Different Types ◽

New System

In this paper, by modifying a known two-dimensional oscillator, we obtain an interesting new oscillator with coexisting limit cycles and point attractors. Then by changing this new system to its forced version and choosing a proper set of parameters, we introduce a chaotic system with some very interesting features. In this system, not only can we see the coexistence of different types of attractors, but also a fascinating phenomenon: some initial conditions can escape from the gravity of nearby attractors and travel far away before being trapped in an attractor beyond the usual access.

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