An algebraic formula for the topological types of one parameter bifurcation diagrams

1989 ◽  
Vol 108 (4) ◽  
pp. 247-265 ◽  
Author(s):  
Takashi Nishimura ◽  
Takuo Fukuda ◽  
Kenji Aoki
Keyword(s):  
Bifurcation Diagrams ◽  
Algebraic Formula ◽  
Parameter Bifurcation

scholarly journals Computational Analysis of Ca2+ Oscillatory Bio-Signals: Two-Parameter Bifurcation Diagrams

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 876
Author(s):  
Wieslaw Marszalek ◽  
Jan Sadecki ◽  
Maciej Walczak
Keyword(s):  
Equilibrium Points ◽  
Autonomous Systems ◽  
Cytosolic Calcium ◽  
Calcium Oscillations ◽  
Dynamical Model ◽  
Bifurcation Diagrams ◽  
Step Size ◽  
Oscillatory Systems ◽  
Two Parameter ◽  
Parameter Bifurcation

Two types of bifurcation diagrams of cytosolic calcium nonlinear oscillatory systems are presented in rectangular areas determined by two slowly varying parameters. Verification of the periodic dynamics in the two-parameter areas requires solving the underlying model a few hundred thousand or a few million times, depending on the assumed resolution of the desired diagrams (color bifurcation figures). One type of diagram shows period-n oscillations, that is, periodic oscillations having n maximum values in one period. The second type of diagram shows frequency distributions in the rectangular areas. Each of those types of diagrams gives different information regarding the analyzed autonomous systems and they complement each other. In some parts of the considered rectangular areas, the analyzed systems may exhibit non-periodic steady-state solutions, i.e., constant (equilibrium points), oscillatory chaotic or unstable solutions. The identification process distinguishes the later types from the former one (periodic). Our bifurcation diagrams complement other possible two-parameter diagrams one may create for the same autonomous systems, for example, the diagrams of Lyapunov exponents, Ls diagrams for mixed-mode oscillations or the 0–1 test for chaos and sample entropy diagrams. Computing our two-parameter bifurcation diagrams in practice and determining the areas of periodicity is based on using an appropriate numerical solver of the underlying mathematical model (system of differential equations) with an adaptive (or constant) step-size of integration, using parallel computations. The case presented in this paper is illustrated by the diagrams for an autonomous dynamical model for cytosolic calcium oscillations, an interesting nonlinear model with three dynamical variables, sixteen parameters and various nonlinear terms of polynomial and rational types. The identified frequency of oscillations may increase or decrease a few hundred times within the assumed range of parameters, which is a rather unusual property. Such a dynamical model of cytosolic calcium oscillations, with mitochondria included, is an important model in which control of the basic functions of cells is achieved through the Ca2+ signal regulation.

ROLE OF PROTECTION IN A TRI-TROPHIC FOOD CHAIN DYNAMICS

2012 ◽  
Vol 20 (02) ◽  
pp. 155-175 ◽  
Author(s):  
S. GAKKHAR ◽  
A. PRIYADARSHI ◽  
SANDIP BANERJEE
Keyword(s):  
Population Density ◽  
Periodic Solutions ◽  
Bifurcation Diagram ◽  
Food Chain ◽  
Stable Equilibrium ◽  
Bifurcation Diagrams ◽  
Chain Dynamics ◽  
Two Parameter ◽  
Parameter Bifurcation

In this paper, the role of protection in stabilizing the tri-trophic food chain dynamics has been explored. The density-dependent protection is provided to bottom prey or middle predator or both. It favors the oscillations damping and has the potential to control the chaotic fluctuations of population density. The bifurcation diagrams have been drawn with respect to protection parameter. They exhibit coexistence of all three species in the form of periodic solutions. The coexistence in the form of stable equilibrium is possible for higher values of protection parameters. Further increase in protection parameters may lead to extinction of one or two species. A two-parameter bifurcation diagram has also been drawn. The Poincaré Maps further confirm the role of protection in controlling the chaos.

Experimental Observations of Propagating Waves and Switching Phenomena in a Coupled Bistable Oscillator System

2014 ◽  
Vol 24 (12) ◽  
pp. 1450157 ◽  
Author(s):  
Kuniyasu Shimizu
Keyword(s):  
Invariant Subspace ◽  
Spectral Distribution ◽  
Numerical Results ◽  
Periodic Waves ◽  
Bifurcation Diagrams ◽  
Propagating Waves ◽  
Different Types ◽  
The Stability ◽  
The One ◽  
Parameter Bifurcation

In this study, we construct a circuit composed of bistable oscillators and we report the experimental observations of quasi-periodic waves propagating in the circuit and compare them with the associated numerical results. Two different types of propagating quasi-periodic waves with identical parameter sets are experimentally verified. The associated numerical results are distinguished by comparing trajectories on the phase planes and by analyzing the one-parameter bifurcation diagrams. Furthermore, the experiments reveal five different types of switching oscillations. The associated numerical results are also presented, and the stability of the switching oscillations (when constrained to an invariant subspace) is numerically investigated. We also calculate spectral distribution for one type of the switching oscillation.

scholarly journals Periodically Pulsed Immunotherapy in a Mathematical Model of Tumor, CD4+ T Cells, and Antitumor Cytokine Interactions

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Hsiu-Chuan Wei ◽  
Jui-Ling Yu ◽  
Chia-Yu Hsu
Keyword(s):  
Mathematical Model ◽  
T Cells ◽  
Immune System ◽  
Interindividual Variability ◽  
Malignant Tumors ◽  
Early Stage ◽  
Bifurcation Diagrams ◽  
Mathematical Analyses ◽  
Cytokine Interactions ◽  
Parameter Bifurcation

Immunotherapy is one of the most recent approaches for controlling and curing malignant tumors. In this paper, we consider a mathematical model of periodically pulsed immunotherapy using CD4+ T cells and an antitumor cytokine. Mathematical analyses are performed to determine the threshold of a successful treatment. The interindividual variability is explored by one-, two-, and three-parameter bifurcation diagrams for a nontreatment case. Numerical simulation conducted in this paper shows that (i) the tumor can be regulated by administering CD4+ T cells alone in a patient with a strong immune system or who has been diagnosed at an early stage, (ii) immunotherapy with a large amount of an antitumor cytokine can boost the immune system to remit or even to suppress tumor cells completely, and (iii) through polytherapy the tumor can be kept at a smaller size with reduced dosages.

scholarly journals Computing Two-Parameter Bifurcation Diagrams for Oscillating Circuits and Systems

IEEE Access ◽  
2019 ◽  
Vol 7 ◽  
pp. 115829-115835 ◽  
Author(s):  
Wieslaw Marszalek ◽  
Helmut Podhaisky ◽  
Jan Sadecki
Keyword(s):  
Bifurcation Diagrams ◽  
Two Parameter ◽  
Parameter Bifurcation

A DESIGN METHOD OF BURSTING USING TWO-PARAMETER BIFURCATION DIAGRAMS IN FITZHUGH–NAGUMO MODEL

2004 ◽  
Vol 14 (07) ◽  
pp. 2241-2252 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI ◽  
KAZUYUKI AIHARA
Keyword(s):  
Mathematical Models ◽  
Bifurcation Diagram ◽  
Bifurcation Analysis ◽  
Design Method ◽  
External Input ◽  
Bifurcation Diagrams ◽  
Parameter Perturbations ◽  
The Brain ◽  
Two Parameter ◽  
Parameter Bifurcation

Spiking and bursting observed in nerve membranes seem to be important when we investigate information representation model in the brain. Many topologically different bursting responses are observed in the mathematical models and their related bifurcation mechanisms have been clarified. In this paper, we propose a design method to generate bursting responses in FitzHugh–Nagumo model with a simple periodic external force based on bifurcation analysis. Some effective parameter perturbations for the amplitude of the external input are given from the two-parameter bifurcation diagram.

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