Bifurcations Involving Fixed Points and Limit Cycles in Biological Systems

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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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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Limit Cycles in Coupled Biological Systems

IFAC Proceedings Volumes ◽

10.1016/s1474-6670(17)63419-0 ◽

1982 ◽

Vol 15 (1) ◽

pp. 643-648

Author(s):

R. Balasubramanian ◽

D.P. Atherton

Keyword(s):

Limit Cycles ◽

Biological Systems

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Exploiting the controlled responses of chaotic elements to design configurable hardware

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2006.1836 ◽

2006 ◽

Vol 364 (1846) ◽

pp. 2483-2494 ◽

Cited By ~ 7

Author(s):

Sudeshna Sinha ◽

William L Ditto

Keyword(s):

Fixed Points ◽

Chaotic System ◽

Limit Cycles ◽

Chaotic Systems ◽

Dynamic Logic ◽

Logic Gates ◽

Threshold Level ◽

Experimental Realization ◽

Configurable Hardware ◽

Simple Manipulation

We discuss how threshold mechanisms can be effectively employed to control chaotic systems onto stable fixed points and limit cycles of widely varying periodicities. Then, we outline the theory and experimental realization of fundamental logic-gates from a chaotic system, using thresholding to effect control. A key feature of this implementation is that a single chaotic ‘processor’ can be flexibly configured (and re-configured) to emulate different fixed or dynamic logic gates through the simple manipulation of a threshold level.

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CHAOTIC BEHAVIOR AND DYNAMICS OF MAPS USED IN A METHOD OF SCRAMBLING SIGNALS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127410028203 ◽

2010 ◽

Vol 20 (12) ◽

pp. 4097-4101

Author(s):

REZA MAZROOEI-SEBDANI ◽

MEHDI DEHGHAN

Keyword(s):

Fixed Points ◽

Limit Cycles ◽

Secure Communication ◽

Chaotic Behavior ◽

Initial Conditions ◽

Chaotic Encryption ◽

Input And Output ◽

Sensitive Dependence ◽

Close Relationship ◽

Existence And Stability

The close relationship between chaos and cryptography makes chaotic encryption a natural candidate for secure communication and cryptography. In this manuscript, we prove that a class of maps that have been proposed as suitable for scrambling signals possess the property of sensitive dependence on initial conditions (s.d.i.c.) necessary for chaos and cryptography. Our result can also be used for generating other maps with s.d.i.c., through a suitable semiconjugacy between their input and output parts. Using the condition of semiconjugacy we also establish for this class of maps rigorous criteria for the existence and stability of their fixed points and limit cycles.

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Spirals and Cycles of Biological Systems via Extended Rosenzweig-MacArthur Model with Ratio-dependent Functional Response

Modern Applied Science ◽

10.5539/mas.v12n9p149 ◽

2018 ◽

Vol 12 (9) ◽

pp. 149

Author(s):

Enobong E. Joshua ◽

Ekemini T. Akpa

Keyword(s):

Numerical Simulations ◽

Functional Response ◽

Limit Cycles ◽

Control Parameter ◽

Exponential Convergence ◽

Biological Systems ◽

Ratio Dependent ◽

Ultimate Bound ◽

Periodic Cycles

This paper investigates stable proper nodes, stable spiral sinks and stable ω−limit cycles of Extended Rosenzweig-MacAthur Model, which incorporates ratio-dependent functional response on predation mechanism. The ultimate bound-edness condition has been used to predict extinction, co-existence, and exponential convergence scenarios of the model. The Poincare-Bendixson results guarantee existence of periodic cycles of the models. The system degenerate from stable spiral sinks to stable ω−limit cycles as control parameter varies. Numerical simulations are provided to support the va-lidity of theoretical findings.

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NONLINEAR OSCILLATIONS (LIMIT CYCLES) IN PHYSICAL AND BIOLOGICAL SYSTEMS

Nonlinear Electromagnetics ◽

10.1016/b978-0-12-709660-5.50016-5 ◽

1980 ◽

pp. 343-389 ◽

Cited By ~ 2

Author(s):

F. Kaiser

Keyword(s):

Limit Cycles ◽

Nonlinear Oscillations ◽

Biological Systems

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Power Laws, Scale Invariance and the Generalized Frobenius Series: Applications to Newtonian and TOV Stars Near Criticality

International Journal of Modern Physics A ◽

10.1142/s0217751x03013892 ◽

2003 ◽

Vol 18 (20) ◽

pp. 3433-3468 ◽

Cited By ~ 14

Author(s):

Matt Visser ◽

Nicolas Yunes

Keyword(s):

Fixed Point ◽

Power Series ◽

Fixed Points ◽

Limit Cycles ◽

Power Laws ◽

Series Solutions ◽

Scale Invariant ◽

Core Mass ◽

Relativistic Case ◽

Power Series Solutions

We present a self-contained formalism for calculating the background solution, the linearized solutions and a class of generalized Frobenius-like solutions to a system of scale-invariant differential equations. We first cast the scale-invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fixed points, we find a second linearized solution, which provides a distinct collection of power laws characterizing the deviations from the fixed point. We prove that generically there will be a region surrounding the fixed point in which the complete general solution can be represented as a generalized Frobenius-like power series with exponents that are integer multiples of the exponents arising in the linearized problem. While discussions of the linearized system are common, and one can often find a discussion of power-series with integer exponents, power series with irrational (indeed complex) exponents are much rarer in the extant literature. The Frobenius-like series we encounter can be viewed as a variant of the rarely-discussed Liapunov expansion theorem (not to be confused with the more commonly encountered Liapunov functions and Liapunov exponents). As specific examples we apply these ideas to Newtonian and relativistic isothermal stars and construct two separate power series with the overlapping radius of convergence. The second of these power series solutions represents an expansion around "spatial infinity," and in realistic models it is this second power series that gives information about the stellar core, and the damped oscillations in core mass and core radius as the central pressure goes to infinity. The power-series solutions we obtain extend classical results; as exemplified for instance by the work of Lane, Emden, and Chandrasekhar in the Newtonian case, and that of Harrison, Thorne, Wakano, and Wheeler in the relativistic case. We also indicate how to extend these ideas to situations where fixed points may not exist — either due to "monotone" flow or due to the presence of limit cycles. Monotone flow generically leads to logarithmic deviations from scaling, while limit cycles generally lead to discrete self-similar solutions.

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