scholarly journals Measuring the criticality of a Hopf bifurcation

2020 ◽  
Vol 101 (4) ◽  
pp. 2541-2549
Author(s):  
Alexei Uteshev ◽  
Tamás Kalmár-Nagy
Keyword(s):  
Vector Field ◽  
Hopf Bifurcation ◽  
Quadratic Function ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Numerical Examples ◽  
Lyapunov Constant ◽  
Parameter Point

Abstract This work is based on the observation that the first Poincaré–Lyapunov constant is a quadratic function of the coefficients of the two-dimensional vector field at a Hopf bifurcation. From a given parameter point, we find the distance to the “Hopf quadric.” This distance provides a measure of the criticality of the Hopf bifurcation. The viability of the approach is demonstrated through numerical examples.

scholarly journals NONORIENTABLE MANIFOLDS IN THREE-DIMENSIONAL VECTOR FIELDS

2003 ◽  
Vol 13 (03) ◽  
pp. 553-570 ◽  
Author(s):  
HINKE M. OSINGA
Keyword(s):  
Vector Field ◽  
Periodic Orbit ◽  
Vector Fields ◽  
Invariant Manifolds ◽  
Three Dimensional ◽  
Period Doubling ◽  
Building Blocks ◽  
Two Dimensional ◽  
Dimensional Vector ◽  
Three Dimensional Vector Field

It is well known that a nonorientable manifold in a three-dimensional vector field is topologically equivalent to a Möbius strip. The most frequently used example is the unstable manifold of a periodic orbit that just lost its stability in a period-doubling bifurcation. However, there are not many explicit studies in the literature in the context of dynamical systems, and so far only qualitative sketches could be given as illustrations. We give an overview of the possible bifurcations in three-dimensional vector fields that create nonorientable manifolds. We mainly focus on nonorientable manifolds of periodic orbits, because they are the key building blocks. This is illustrated with invariant manifolds of three-dimensional vector fields that arise from applications. These manifolds were computed with a new algorithm for computing two-dimensional manifolds.

scholarly journals Bifurcations of Tumor-Immune Competition Systems with Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Ping Bi ◽  
Heying Xiao
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Codimension Two ◽  
Codimension One ◽  
Distribution Of Eigenvalues ◽  
Steady State Bifurcation

A tumor-immune competition model with delay is considered, which consists of two-dimensional nonlinear differential equation. The conditions for the linear stability of the equilibria are obtained by analyzing the distribution of eigenvalues. General formulas for the direction, period, and stability of the bifurcated periodic solutions are given for codimension one and codimension two bifurcations, including Hopf bifurcation, steady-state bifurcation, and B-T bifurcation. Numerical examples and simulations are given to illustrate the bifurcations analysis and obtained results.

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