scholarly journals On the 3-dimensional Hopf bifurcation via averaging theory of third order

2017 ◽  
Vol 41 ◽  
pp. 1053-1071
Author(s):  
Elouahma BENDIB ◽  
Sabrina BADI ◽  
Amar MAKHLOUF
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
Third Order ◽  
3 Dimensional

scholarly journals On the Zero-Hopf Bifurcation of the Lotka–Volterra Systems in R 3

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1137
Author(s):  
Maoan Han ◽  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Periodic Orbits ◽  
Differential System ◽  
Equilibrium Points ◽  
Averaging Theory ◽  
Third Order ◽  
Differential Systems ◽  
3 Dimensional ◽  
Volterra Systems

Here we study 3-dimensional Lotka–Volterra systems. It is known that some of these differential systems can have at least four periodic orbits bifurcating from one of their equilibrium points. Here we prove that there are some of these differential systems exhibiting at least six periodic orbits bifurcating from one of their equilibrium points. We remark that these systems with such six periodic orbits are non-competitive Lotka–Volterra systems. The proof is done using the algorithm that we provide for computing the periodic solutions that bifurcate from a zero-Hopf equilibrium based in the averaging theory of third order. This algorithm can be applied to any differential system having a zero-Hopf equilibrium.

scholarly journals 3-dimensional Hopf bifurcation via averaging theory

2007 ◽  
Vol 17 (3) ◽  
pp. 529-540 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Claudio A. Buzzi ◽  
Paulo R. da Silva ◽  
Keyword(s):  
Hopf Bifurcation ◽  
Averaging Theory ◽  
3 Dimensional

scholarly journals $3$ - dimensional Hopf bifurcation via averaging theory of second order

2009 ◽  
Vol 25 (4) ◽  
pp. 1287-1295 ◽  
Author(s):  
Jaume Llibre ◽  
◽  
Amar Makhlouf ◽  
Sabrina Badi ◽  
◽  
...  
Keyword(s):  
Hopf Bifurcation ◽  
Second Order ◽  
Averaging Theory ◽  
3 Dimensional

Characterization of Extreme Points of Multi-Stochastic Tensors

2016 ◽  
Vol 16 (3) ◽  
pp. 459-474 ◽  
Author(s):  
Rihuan Ke ◽  
Wen Li ◽  
Mingqing Xiao
Keyword(s):  
Operation Research ◽  
Convex Combination ◽  
Sufficient Conditions ◽  
Extreme Points ◽  
Stochastic Matrices ◽  
Third Order ◽  
New Approach ◽  
3 Dimensional ◽  
Necessary And Sufficient

AbstractStochastic matrices play an important role in the study of probability theory and statistics, and are often used in a variety of modeling problems in economics, biology and operation research. Recently, the study of tensors and their applications became a hot topic in numerical analysis and optimization. In this paper, we focus on studying stochastic tensors and, in particular, we study the extreme points of a set of multi-stochastic tensors. Two necessary and sufficient conditions for a multi-stochastic tensor to be an extreme point are established. These conditions characterize the “generators” of multi-stochastic tensors. An algorithm to search the convex combination of extreme points for an arbitrary given multi-stochastic tensor is developed. Based on our obtained results, some expression properties for third-order and n-dimensional multi-stochastic tensors (${n=3}$ and 4) are derived, and all extreme points of 3-dimensional and 4-dimensional triply-stochastic tensors can be produced in a simple way. As an application, a new approach for the partially filled square problem under the framework of multi-stochastic tensors is given.

Hopf bifurcation of the third-order Hénon system based on an explicit criterion

2009 ◽  
Vol 70 (9) ◽  
pp. 3227-3235 ◽  
Author(s):  
Enying Li ◽  
Guangyao Li ◽  
Guilin Wen ◽  
Hu Wang
Keyword(s):  
Hopf Bifurcation ◽  
Explicit Criterion ◽  
Third Order ◽  
The Third

scholarly journals An Approach to Robust Control of the Hopf Bifurcation

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Giacomo Innocenti ◽  
Roberto Genesio ◽  
Alberto Tesi
Keyword(s):  
Hopf Bifurcation ◽  
State Feedback ◽  
Center Manifold ◽  
State Feedback Control ◽  
Control Input ◽  
Nonlinear Transformations ◽  
Third Order ◽  
Novel Approach ◽  
Linear And Nonlinear ◽  
Nonlinear State

The paper illustrates a novel approach to modify the Hopf bifurcation nature via a nonlinear state feedback control, which leaves the equilibrium properties unchanged. This result is achieved by recurring to linear and nonlinear transformations, which lead the system to locally assume the ordinary differential equation representation. Third-order models are considered, since they can be seen as proper representatives of a larger class of systems. The explicit relationship between the control input and the Hopf bifurcation nature is obtained via a frequency approach, that does not need the computation of the center manifold.

Third-Order Memristive Morris–Lecar Model of Barnacle Muscle Fiber

2017 ◽  
Vol 27 (04) ◽  
pp. 1730015 ◽  
Author(s):  
Vetriveeran Rajamani ◽  
Maheshwar PD. Sah ◽  
Zubaer Ibna Mannan ◽  
Hyongsuk Kim ◽  
Leon Chua
Keyword(s):  
Hopf Bifurcation ◽  
Muscle Fiber ◽  
Limit Cycle Oscillation ◽  
Potassium Ions ◽  
Stable Equilibrium Point ◽  
Third Order ◽  
The Third ◽  
Barnacle Muscle ◽  
Subcritical Hopf Bifurcation ◽  
Barnacle Muscle Fiber

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris–Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris–Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris–Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris–Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris–Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon.

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