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Continuous Models Using Ordinary Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
A Hopf bifurcation is a local bifurcation, which occurs when a complex conjugate pair of eigenvalues λ± of the linearized system about the equilibrium point crosses from Re(λ±)<0 (the left half-plane) to the Re(λ±)>0 (the right half-plane) and a limit cycle emerges from the fixed point. If the limit cycle that emerges is stable, we call it a supercritical Hopf bifurcation and the emergence of an unstable limit cycle is called a subcritical Hopf bifurcation. Thus, a Hopf bifurcation occurs when we have a change in stability of a fixed point from one type of focus to another. Along with the change in stability, a limit cycle emerges in the phase plane.
Modeling Virus Dynamics in Time and Space
Published in Ranjit Kumar Upadhyay, Satteluri R. K. Iyengar, Spatial Dynamics and Pattern Formation in Biological Populations, 2021
Ranjit Kumar Upadhyay, Satteluri R. K. Iyengar
where h(I)=aI/(1+bI). Using the bifurcation theory, the authors have shown that the temporal model system exhibits backward bifurcation, Hopf bifurcation, and Bogdanov – Takens bifurcation. They suggested that to eradicate the disease, we should raise the efficiency and enlarge the capacity of treatment. That is, we should improve our medical technology and invest in medicines, beds, and so on to give patients timely treatment.
Reactor Steady-State Multiplicity and Stability
Published in James J. Carberry, Arvind Varma, Chemical Reaction and Reactor Engineering, 2020
Massimo Morbidelli, Arvind Varma, Rutherford Aris
The determination of the Hopf bifurcation points is only the first step toward defining the dynamic behavior of this system, whose detailed understanding requires a detailed analysis of each periodic branch. This involves an identification of the structure of the system dynamics as the bifurcation parameter changes away from the Hopf bifurcation point. Analysis of this type usually requires tedious integration of the transient model equations. Doedel and Heinemann (1983) have applied a continuation method for the calculation of the entire branch of periodic solutions, both stable and unstable. This method is based on a transformation of the original initial value problem into a boundary value problem, where the period is taken as unknown and correspondingly the periodicity condition is added. The final equations are then discretized using orthogonal collocation, and the resulting system of nonlinear algebraic equations is solved through Newton’s method. This procedure has been implemented by Doedel (1981) on a computer code which also provides the steady-state solution branches using Keller’s (1977) arc-length method for avoiding singularity problems in the Jacobian at the static bifurcation point, as well as the Hopf bifurcation points and the bifurcating periodic solution, using the usual perturbation techniques. This appears to be quite a useful tool for the complete study of the dynamics of a given system, since it gives the entire bifurcation diagram (i.e., both steady state and periodic solution branches, both stable and unstable). The stability character of periodic solutions is determined through the corresponding Floquet multipliers.
On optimal harvesting policy for two economically beneficial species mysida and herring: a clue for conservation biologist through mathematical model
Published in International Journal of Modelling and Simulation, 2023
Prahlad Majumdar, Sabyasachi Bhattacharya, Susmita Sarkar, Uttam Ghosh
The Hopf bifurcation happens when an equilibrium point changes its stability letting the existence of the closed orbit around it [30]. The system (6) will experience Hopf bifurcation at the equilibrium point , if both roots of the equation (11) are purely imaginary, which is possible if and the transversality condition is satisfied [19,22]. For this purpose, we consider the intrinsic growth rate () as the bifurcation parameter and determine its critical value , for which , and the condition of transversality i.e. is satisfied.
Nonlinear and bifurcation analysis for a novel heterogeneous continuum model and numerical tests
Published in Transportmetrica B: Transport Dynamics, 2022
Weilin Ren, Rongjun Cheng, Hongxia Ge
In this section, we theoretically study the Hopf bifurcation behaviour of traffic flow to provide a theoretical basis for subsequent numerical simulations. This section consists of three parts. The first part (Section 5.1) is to discuss the types of equilibrium points and their stability. The second part (Section 5.2) is to derive the condition for the existence of Hopf bifurcation. The third part (Section 5.3) is to judge the type of Hopf bifurcation, subcritical or supercritical one. Subcritical Hopf bifurcation leads to unstable limit cycle, while supercritical one brings about stable limit cycle.
Bifurcation Analysis of Spatial Xenon Oscillations in Large Pressurized Heavy Water Reactors Using Multipoint Reactor Kinetics with Thermal-Hydraulic Feedback
Published in Nuclear Science and Engineering, 2021
Abhishek Chakraborty, Suneet Singh, M. P. S. Fernando
In subcritical Hopf bifurcation, an unstable limit cycle bifurcates from a stable fixed point. Any perturbation with amplitude less than that of the limit cycle causes neighboring trajectories to be attracted to the stable fixed point. Any perturbation with amplitude higher than that of the limit cycle also repels neighboring trajectories from the unstable limit cycle. Hence, in a subcritical condition after bifurcation, the trajectories must jump to a distant attractor.