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Bifurcation Behaviour of Non-Linear Systems
Published in T. Thyagarajan, D. Kalpana, Linear and Non-Linear System Theory, 2020
The saddle node bifurcation is a collision and disappearance of two equilibria in dynamical systems. This occurs when the critical equilibrium has one zero eigen value. This phenomenon is also called ‘fold’ or ‘limit point’ or ‘tangent’ or ‘blue sky’ bifurcation.
How random fluctuations can generate and suppress complex oscillatory regimes in continuous stirred tank reactors
Published in Combustion Theory and Modelling, 2023
Irina Bashkirtseva, Lev Ryashko
In Figure 1, we show extreme values of x- and y-coordinates of attractors, blue for limit cycles and red for stable equilibria, in the interval . Here, limit cycles are observed in the parameter range while the stable equilibria are seen for , where . Here, marks the subcritical Hopf bifurcation, and is the saddle-node bifurcation point. So, in the parameter range , the system (1) is bistable and exhibits coexistence of the stable equilibrium and limit cycle. Thus, for , the thermochemical system can be both in equilibrium and oscillatory mode in dependence on the initial state.
You can create your own bifurcation
Published in International Journal of Mathematical Education in Science and Technology, 2021
A necessary condition for an equilibrium point of exhibiting a bifurcation is at a critical value of μ. For , , and and experience a transcritical bifurcation (Argyris et al., 1994; Guckenheimer & Holmes, 2002; Kuznetsov, 2004; Nayfeh & Balachandran, 1995; Strogatz, 2015; Wiggins, 1990), because these points exchange their stabilities at the critical number . For , and the four equilibrium points are involved in the bifurcation shown in Figure 1. In this bifurcation diagram in the plane , solid line represents asymptotically stable steady-state; dashed line, unstable steady-state. Observe that, at , and are created with opposite stabilities, and and switch their stabilities. Thus, a saddle-node bifurcation and a transcritical bifurcation simultaneously occur at .
The bifurcation analysis of an SIRS epidemic model with immunity age and constant treatment
Published in Applicable Analysis, 2021
Hui Cao, Xiaoyan Gao, Jianquan Li, Dongxue Yan, Zongmin Yue
For model (4) with conditions (5) and (6), our aim is to investigate the influence of the shortest immune time on the spread of the disease, and the existence of saddle-node bifurcation, Bogdanov–Takens bifurcation, and Hopf bifurcation by using the bifurcation theory. as an abstract non-densely defined Cauchy problem and derive the conditions for the existence of all the feasible equilibrium of system. Through analyzing the location of eigenvalues of the associated characteristic equation, we investigate the stabilities of all the feasible equilibria of the system under certain conditions. In addition, the criterion for existence of Hopf bifurcation around an endemic equilibrium is obtained.