scholarly journals Bifurcations of Oscillatory and Rotational Solutions of Double Pendulum with Parametric Vertical Excitation

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Michal Marszal ◽  
Krzysztof Jankowski ◽  
Przemyslaw Perlikowski ◽  
Tomasz Kapitaniak
Keyword(s):  
Periodic Solutions ◽  
Parameter Space ◽  
Double Pendulum ◽  
Bifurcation Diagrams ◽  
Kinematic Excitation ◽  
Vertical Excitation ◽  
Two Parameter ◽  
Domain Of Periodic Solutions

This paper investigates dynamics of double pendulum subjected to vertical parametric kinematic excitation. It includes detailed bifurcation diagrams in two-parameter space (excitation’s frequency and amplitude) for both oscillations and rotations in the domain of periodic solutions.

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.

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.

Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields

2015 ◽  
Vol 25 (11) ◽  
pp. 1550150 ◽  
Author(s):  
Oxana Cerba Diaconescu ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Parameter Space ◽  
Group Action ◽  
Sufficient Conditions ◽  
Phase Portraits ◽  
Topological Spaces ◽  
Canonical Forms ◽  
Affine Transformations ◽  
Bifurcation Diagrams ◽  
Dimensional Parameter ◽  
Quotient Spaces

In this article, we consider the class [Formula: see text] of all real quadratic differential systems [Formula: see text], [Formula: see text] with gcd (p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities. We first construct compactified canonical forms for the class [Formula: see text] so as to include limit points in the 12-dimensional parameter space of this class. We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and we place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under this group action. We also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.

Hopf Bifurcation of a Type of Neuron Model with Multiple Time Delays

2019 ◽  
Vol 29 (12) ◽  
pp. 1950163 ◽  
Author(s):  
Suqi Ma
Keyword(s):  
Hopf Bifurcation ◽  
Parameter Space ◽  
Time Delays ◽  
Neuron Model ◽  
Multiple Time ◽  
Multiple Time Delays ◽  
The Stability ◽  
Geometrical Scheme ◽  
Delay Differential ◽  
Two Parameter

By applying a geometrical scheme developed to tackle the eigenvalue problem of delay differential equations with multiple time delays, Hopf bifurcation of Hopfield neuron model is analyzed in two-parameter space. By the introduction of two new angles, the calculation of imaginary roots is carried out analytically and effectively. By increasing the parameter to cross over the Hopf bifurcation lines, the stability switching direction is confirmed. The method is a useful tool to show the partition of stable and unstable regions in two-parameter space and detect double Hopf bifurcation further. The typified dynamical behaviors based on nearby double Hopf points are analyzed by applying the normal form technique and center manifold method.

Instabilities of exact, time-periodic solutions of the incompressible Euler equations

2000 ◽  
Vol 404 ◽  
pp. 269-287 ◽  
Author(s):  
JOSEPH A. BIELLO ◽  
KENNETH I. SALDANHA ◽  
NORMAN R. LEBOVITZ
Keyword(s):  
Periodic Solutions ◽  
Linear Stability ◽  
Euler Equations ◽  
Parameter Space ◽  
Spatial Scales ◽  
Inviscid Flow ◽  
Steady Flows ◽  
Periodic Flows ◽  
Finite Dimensional ◽  
Time Periodic Solutions

We consider the linear stability of exact, temporally periodic solutions of the Euler equations of incompressible, inviscid flow in an ellipsoidal domain. The problem of linear stability is reduced, without approximation, to a hierarchy of finite-dimensional Floquet problems governing fluid-dynamical perturbations of differing spatial scales and symmetries. We study two of these Floquet problems in detail, emphasizing parameter regimes of special physical significance. One of these regimes includes periodic flows differing only slightly from steady flows. Another includes long-period flows representing the nonlinear outcome of an instability of steady flows. In both cases much of the parameter space corresponds to instability, excepting a region adjacent to the spherical configuration. In the second case, even if the ellipsoid departs only moderately from a sphere, there are filamentary regions of instability in the parameter space. We relate this and other features of our results to properties of reversible and Hamiltonian systems, and compare our results with related studies of periodic flows.

BUILDING ELECTRONIC BURSTERS WITH THE MORRIS–LECAR NEURON MODEL

2006 ◽  
Vol 16 (12) ◽  
pp. 3617-3630 ◽  
Author(s):  
ALEXANDRE WAGEMAKERS ◽  
MIGUEL A. F. SANJUÁN ◽  
JOSÉ M. CASADO ◽  
KAZUYUKI AIHARA
Keyword(s):  
Phase Space ◽  
Parameter Space ◽  
Neuron Model ◽  
Electronic Circuit ◽  
Initial Point ◽  
Bifurcation Diagrams ◽  
Design And Implementation ◽  
Bursting Neurons ◽  
Fast Spiking ◽  
Dynamical Features

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models.

Control of Bifurcation and Chaos in a Class of Digital Tanlock Loops with Modified Loop Structure

2014 ◽  
Vol 24 (04) ◽  
pp. 1430014
Author(s):  
Bishnu Charan Sarkar ◽  
Saumendra Sankar De Sarkar ◽  
Tanmoy Banerjee
Keyword(s):  
Analytical Study ◽  
Nonlinear Behavior ◽  
Convergence Time ◽  
Loop Structure ◽  
Structure Parameter ◽  
Bifurcation Diagrams ◽  
Bifurcation And Chaos ◽  
Time Delayed Feedback ◽  
Large Frequency ◽  
Two Parameter

A detailed parametric space study of the nonlinear behavior of a kind of Digital Tanlock Loops (DTL) incorporating a modified loop structure using a time delayed feedback technique has been studied in this paper. The analytical study reveals that the modified loop shows better performance compared to a conventional DTL (CDTL). The superiority of the modified DTL (MDTL) has been established by a numerical simulation study. Two parameter bifurcation diagrams, supported by Lyapunov exponent spectrums, have been used to give a detailed idea about the nonlinear dynamics of the loop for a wide parameter range. The MDTL shows large frequency acquisition range (FAR) and faster convergence time (CT) than a CDTL for suitably chosen values of the design parameter. An estimate for the optimum value of the structure parameter for quick convergence has also been presented in the present study.

Instantaneous Kinematics of a Tangent-Plane in Two-Parameter Space Motion

1994 ◽  
Vol 116 (1) ◽  
pp. 210-214
Author(s):  
D. P. Sathyadev ◽  
A. H. Soni
Keyword(s):  
Parameter Space ◽  
Tangent Plane ◽  
Plane Motion ◽  
Parameter Family ◽  
Spherical Image ◽  
Space Motion ◽  
Two Parameter

A tangent-plane undergoing two-parameter motion envelopes a surface called the tangent-plane envelope. Such surfaces can be considered as the envelope of a two-parameter family of planes or ∞2 family of planes. The properties of the tangent-plane motion are characterized through the properties of the spherical image of the normal to the surface it envelopes. This paper presents a methodology to locate a family of planes that envelope surfaces with similar characteristics.

Nonlinear Dynamics of a Pendulum Excited by a Crank-Shaft-Slider Mechanism

2014 ◽  
Author(s):  
Rafael H. Avanço ◽  
Hélio A. Navarro ◽  
Reyolando M. L. R. F. Brasil ◽  
José M. Balthazar
Keyword(s):  
Nonlinear Dynamics ◽  
Lyapunov Exponents ◽  
Nonlinear Model ◽  
Special Interest ◽  
Initial Conditions ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Numerical Techniques ◽  
Vertical Excitation ◽  
Time Histories

In this analysis, we consider the dynamics of a pendulum under vertical excitation of a crank-shaft-slider mechanism. The nonlinear model approaches that of a classical parametrically excited pendulum when the ratio of the length of the shaft to the radius of the crank is very large. Numerical techniques are employed to investigate the results for different parameters and initial conditions. Lyapunov exponents, bifurcation diagrams, time histories and phase portraits are presented to explore conditions when the pendulum performs or not full rotations. Of special interest are the resonance regions. Rotations together with oscillations and chaos were observed in some resonance zones.

LIMIT CYCLES IN 3D LOTKA–VOLTERRA SYSTEMS APPEARING AFTER PERTURBATION OF HOPF CENTER

2008 ◽  
Vol 18 (12) ◽  
pp. 3647-3656 ◽  
Author(s):  
Ł. J. GOŁASZEWSKI ◽  
P. SŁAWIŃSKI ◽  
H. ŻOŁADEK
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Limit Cycle ◽  
Limit Cycles ◽  
Invariant Torus ◽  
Small Amplitude ◽  
Parameter Family ◽  
Volterra Systems ◽  
Parameter Perturbations ◽  
Two Parameter

We study the system ẋ = x(y+2z+(15/2η2)u), ẏ = y(x-2z-(7/2η2)u), ż = -z(x+y+(4/η2)u), u = x+y+z-1, and its two-parameter perturbations. We show that before perturbation there exists a one-parameter family of periodic solutions obtained via a nondegenarate Hopf bifurcation and after perturbation there remains at most one limit cycle of small amplitude and bounded period. Moreover, we found that a secondary Hopf bifurcation to an invariant torus occurs after the perturbation.

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