Complex eigenvalues of an arbitrary map (dots). In case of the Hopf bifurcation, two complex conjugate eigenvalues cross the imaginary axis.
In the mathematical theory of bifurcations, a Hopfbifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic solution arises.[1] More accurately, it is a local bifurcation in which a fixed point of a dynamical system loses stability, as a pair of complex conjugate eigenvalues—of the linearization around the fixed point—crosses the complex plane imaginary axis as a parameter crosses a threshold value. Under reasonably generic assumptions about the dynamical system, the fixed point becomes a small-amplitude limit cycle as the parameter changes.
A Hopf bifurcation is also known as a Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Aleksandr Andronov and Eberhard Hopf.
In the mathematical theory of bifurcations, a Hopfbifurcation is a critical point where, as a parameter changes, a system's stability switches and a periodic...
for by the bifurcation theory. As the carrying capacity increases, the equilibrium of the dynamical system becomes unstable. The bifurcation can be obtained...
Hopfbifurcation, the limit cycle should have size ∝ ε 1 / 2 , {\displaystyle \propto \varepsilon ^{1/2},} we may attempt to convert this to a Hopf bifurcation...
In bifurcation theory, a field within mathematics, a pitchfork bifurcation is a particular type of local bifurcation where the system transitions from...
variables. Another example shows how a center manifold models the Hopfbifurcation that occurs for parameter a ≈ 4 {\displaystyle a\approx 4} in the delay...
increasing rate Hopfbifurcation, in mathematics, a local bifurcation; when the first Lyapunov coefficient is negative, the bifurcation is called supercritical...
Eberhard Frederich Ferdinand Hopf (April 4, 1902 in Salzburg, Austria-Hungary – July 24, 1983 in Bloomington, Indiana, USA) was a German mathematician...
context of a biological system, bifurcation theory describes how small changes in an input parameter can cause a bifurcation or qualitative change in the...
bifurcations called a Hopfbifurcation and an infinite period bifurcation.[citation needed] Biological applications of bifurcation theory Biophysics Biostatistics...
considered in the literature. Another explanation is that speed wobble is a Hopfbifurcation, whereby a system changes from one state (a stable ride) to a second...
Examples of limit cycles branching from fixed points near Hopfbifurcation. Trajectories in red, stable structures in dark blue, unstable structures in...
JSTOR 2461358. S2CID 84799770. Shen, Zuolin; Wei, Junjie (2018-06-01). "Hopfbifurcation analysis in a diffusive predator-prey system with delay and surplus...
S2CID 213816027. Gori, Luca; Guerrini, Luca; Sodini, Mauro (2014). "HopfBifurcation in a Cobweb Model with Discrete Time Delays". Discrete Dynamics in...
body inertia on the nonlinear wheel-set hunting recognized by the hopfbifurcation theory". Int J Auto Engng: 186–196. Iwnicki, Simon (2006). Handbook...
value, both equilibrium points lose stability through a subcritical Hopfbifurcation. When ρ = 28, σ = 10, and β = 8/3, the Lorenz system has chaotic solutions...
damped out, but may be amplified. Mathematically, this corresponds to a Hopfbifurcation where increasing one of the parameters beyond a certain value leads...
where pulsation sets in during the star's evolution correspond to a Hopfbifurcation. The existence of a center manifold eliminates the possibility of chaotic...
to the approach to a saddle-node bifurcation, a subcritical Hopfbifurcation, or an inverse period-doubling bifurcation. In the apparently periodic phases...
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