scholarly journals Codimension-Two Bifurcations of Fixed Points in a Class of Discrete Prey-Predator Systems

2011 ◽  
Vol 2011 ◽  
pp. 1-27 ◽  
Author(s):  
R. Khoshsiar Ghaziani ◽  
W. Govaerts ◽  
C. Sonck
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Dynamic Behaviour ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Volterra System ◽  
Codimension Two ◽  
Bifurcation Points ◽  
Two Parameters ◽  
Form Stability

The dynamic behaviour of a Lotka-Volterra system, described by a planar map, is analytically and numerically investigated. We derive analytical conditions for stability and bifurcation of the fixed points of the system and compute analytically the normal form coefficients for the codimension 1 bifurcation points (flip and Neimark-Sacker), and so establish sub- or supercriticality of these bifurcation points. Furthermore, by using numerical continuation methods, we compute bifurcation curves of fixed points and cycles with periods up to 16 under variation of one and two parameters, and compute all codimension 1 and codimension 2 bifurcations on the corresponding curves. For the bifurcation points, we compute the corresponding normal form coefficients. These quantities enable us to compute curves of codimension 1 bifurcations that branch off from the detected codimension 2 bifurcation points. These curves form stability boundaries of various types of cycles which emerge around codimension 1 and 2 bifurcation points. Numerical simulations confirm our results and reveal further complex dynamical behaviours.

scholarly journals Dynamics and bifurcations of a ratio-dependent predator-prey model

2022 ◽  
Vol 40 ◽  
pp. 1-20
Author(s):  
Parisa Azizi ◽  
Reza Khoshsiar Ghaziani
Keyword(s):  
Normal Form ◽  
Limit Cycles ◽  
Numerical Continuation ◽  
Predator Prey ◽  
Bifurcation Points ◽  
Predator Prey Model ◽  
Prey Model ◽  
Ratio Dependent

In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.

A GALLERY OF OSCILLATIONS IN A RESONANT ELECTRIC CIRCUIT: HOPF-HOPF AND FOLD-FLIP INTERACTIONS

2008 ◽  
Vol 18 (02) ◽  
pp. 481-494 ◽  
Author(s):  
GUSTAVO REVEL ◽  
DIEGO M. ALONSO ◽  
JORGE L. MOIOLA
Keyword(s):  
Hopf Bifurcation ◽  
Normal Form ◽  
Chaotic Motion ◽  
Electric Circuit ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Bifurcation Diagrams ◽  
Frequency Components ◽  
Truncated Normal ◽  
2D Torus

In this work, the dynamics of a coupled electric circuit is studied. Several bifurcation diagrams associated with the truncated normal form of the Hopf-Hopf bifurcation are presented. The bifurcation curves are obtained by numerical continuation methods. The existence of quasi-periodic solutions with two (2D torus) and three (3D torus) frequency components is shown. These, in certain way, are close (or have a tendency to end up) to chaotic motion. Furthermore, two fold-flip bifurcations are detected in the vicinity of the Hopf-Hopf bifurcation, and are classified correspondingly. The analysis is completed with time simulations, the continuation of several limit cycle bifurcations and the indication of resonance points.

Codimension-One and -Two Bifurcations of a Three-Dimensional Discrete Game Model

2021 ◽  
Vol 31 (02) ◽  
pp. 2150023
Author(s):  
Zohreh Eskandari ◽  
Javad Alidousti ◽  
Reza Khoshsiar Ghaziani
Keyword(s):  
Center Manifold ◽  
Continuation Method ◽  
Three Dimensional ◽  
Period Doubling ◽  
Numerical Continuation ◽  
Game Model ◽  
Codimension Two ◽  
Codimension One ◽  
Two Parameters ◽  
Discrete Game

In this paper, bifurcation analysis of a three-dimensional discrete game model is provided. Possible codimension-one (codim-1) and codimension-two (codim-2) bifurcations of this model and its iterations are investigated under variation of one and two parameters, respectively. For each bifurcation, normal form coefficients are calculated through reduction of the system to the associated center manifold. The bifurcations detected in this paper include transcritical, fold, flip (period-doubling), Neimark–Sacker, period-doubling Neimark–Sacker, resonance 1:2, resonance 1:3, resonance 1:4 and fold-flip bifurcations. Moreover, we depict bifurcation diagrams corresponding to each bifurcation with the aid of numerical continuation method. These bifurcation curves not only confirm our analytical results, but also reveal a richer dynamics of the model especially in the higher iterations.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

scholarly journals Stationary localized structures and the effect of the delayed feedback in the Brusselator model

2018 ◽  
Vol 376 (2135) ◽  
pp. 20170385 ◽  
Author(s):  
B. Kostet ◽  
M. Tlidi ◽  
F. Tabbert ◽  
T. Frohoff-Hülsmann ◽  
S. V. Gurevich ◽  
...  
Keyword(s):  
Reaction Diffusion ◽  
Dissipative Structures ◽  
Delayed Feedback ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Delayed Feedback Control ◽  
Brusselator Model ◽  
Localized Structures ◽  
Reaction Diffusion Model ◽  
Spatial Dimensions

The Brusselator reaction–diffusion model is a paradigm for the understanding of dissipative structures in systems out of equilibrium. In the first part of this paper, we investigate the formation of stationary localized structures in the Brusselator model. By using numerical continuation methods in two spatial dimensions, we establish a bifurcation diagram showing the emergence of localized spots. We characterize the transition from a single spot to an extended pattern in the form of squares. In the second part, we incorporate delayed feedback control and show that delayed feedback can induce a spontaneous motion of both localized and periodic dissipative structures. We characterize this motion by estimating the threshold and the velocity of the moving dissipative structures. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.

scholarly journals Bifurcations from steady to quasi-periodic flows in a laterally heated cavity filled with low Prandtl number fluids

2018 ◽  
Vol 861 ◽  
pp. 223-252 ◽  
Author(s):  
A. Medelfef ◽  
D. Henry ◽  
A. Bouabdallah ◽  
S. Kaddeche
Keyword(s):  
Prandtl Number ◽  
Liquid Metals ◽  
Three Dimensional ◽  
Numerical Continuation ◽  
Arnoldi Method ◽  
Periodic Flows ◽  
Bifurcation Points ◽  
Prandtl Numbers ◽  
Stable Flow ◽  
Flow States

This study deals with the transition toward quasi-periodicity of buoyant convection generated by a horizontal temperature gradient in a three-dimensional parallelepipedic cavity with dimensions $4\times 2\times 1$ (length $\times$ width $\times$ height). Numerical continuation techniques, coupled with an Arnoldi method, are used to locate the steady and Hopf bifurcation points as well as the different steady and periodic flow branches emerging from them for Prandtl numbers ranging from 0 to 0.025 (liquid metals). Our results highlight the existence of two steady states along with many periodic cycles, all with different symmetries. The bifurcation scenarios consist of complex paths between these different solutions, giving a succession of stable flow states as the Grashof number is increased, from steady to periodic and quasi-periodic. The change of these scenarios with the Prandtl number, in connection with the crossing of bifurcation points, was carefully analysed.

Spatiotemporal Dynamics in Reaction–Diffusion Neural Networks Near a Turing–Hopf Bifurcation Point

2019 ◽  
Vol 29 (11) ◽  
pp. 1950154 ◽  
Author(s):  
Jiazhe Lin ◽  
Rui Xu ◽  
Xiaohong Tian
Keyword(s):  
Neural Networks ◽  
Hopf Bifurcation ◽  
Normal Form ◽  
Bifurcation Point ◽  
Reaction Diffusion ◽  
Spatiotemporal Dynamics ◽  
Normal Form Theory ◽  
Hopf Bifurcation Point ◽  
Codimension Two ◽  
Form Theory

Since the electromagnetic field of neural networks is heterogeneous, the diffusion phenomenon of electrons exists inevitably. In this paper, we investigate the existence of Turing–Hopf bifurcation in a reaction–diffusion neural network. By the normal form theory for partial differential equations, we calculate the normal form on the center manifold associated with codimension-two Turing–Hopf bifurcation, which helps us understand and classify the spatiotemporal dynamics close to the Turing–Hopf bifurcation point. Numerical simulations show that the spatiotemporal dynamics in the neighborhood of the bifurcation point can be divided into six cases and spatially inhomogeneous periodic solution appears in one of them.

scholarly journals Targeted Energy Transfers for Suppressing Regenerative Machine Tool Vibrations

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
Amir Nankali ◽  
Young S. Lee ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Machine Tool ◽  
Passive Control ◽  
Stability Margin ◽  
Numerical Continuation ◽  
Continuation Methods ◽  
Transition Curve ◽  
Hopf Bifurcation Point ◽  
Energy Transfers ◽  
Tool Vibrations ◽  
Nonlinear Energy

We study the dynamics of targeted energy transfers in suppressing chatter instability in a single-degree-of-freedom (SDOF) machine tool system. The nonlinear regenerative (time-delayed) cutting force is a main source of machine tool vibrations (chatter). We introduce an ungrounded nonlinear energy sink (NES) coupled to the tool, by which energy transfers from the tool to the NES and efficient dissipation can be realized during chatter. Studying variations of a transition curve with respect to the NES parameters, we analytically show that the location of the Hopf bifurcation point is influenced only by the NES mass and damping coefficient. We demonstrate that application of a well-designed NES renders the subcritical limit cycle oscillations (LCOs) into supercritical ones, followed by Neimark–Sacker and saddle-node bifurcations, which help to increase the stability margin in machining. Numerical and asymptotic bifurcation analyses are performed and three suppression mechanisms are identified. The asymptotic stability analysis is performed to study the domains of attraction for these suppression mechanisms which exhibit good agreement with the bifurcations sets obtained from the numerical continuation methods. The results will help to design nonlinear energy sinks for passive control of regenerative instabilities in machining.

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