🕸Systems Engineering: Bifurcation (Definitions)

"A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed. This occurs when an attractor in the system changes in response to change in the value of a parameter." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"A qualitative change in a system’s dynamics as a parameter is varied. One-parameter bifurcation diagrams often depict invariant sets of the dynamics against a single parameter, indicating stability and any bifurcation points. Two-parameter bifurcation diagrams depict curves in a parameter plane on which one-parameter bifurcations occur." (Alan Champneys, "Dynamics of Parametric Excitation", 2009) 

"A qualitative change in the essential behavior of a mathematical system (e. g., algebraic equations, iterated maps, differential equations), as a parameter is varied. Usually involves an abrupt appearance and/or disappearance of solutions." (Oded Regev, "Chaos and Complexity in Astrophysics", 2009) 

"Bifurcation is a qualitative change of the phase portrait." (George Osipenko, "Center Manifolds", 2009) 

"In parametrized dynamical systems a bifurcation occurs when a qualitative change is invoked by a change of parameters. In models such a qualitative change corresponds to transition between dynamical regimes. In the generic theory a finite list of cases is obtained, containing elements like ‘saddle-node’, ‘period doubling’, ‘Hopf bifurcation’ and many others." (Henk W Broer & Heinz Hanssmann, "Hamiltonian Perturbation Theory (and Transition to Chaos)", 2009) 

"In mathematical models, a bifurcation occurs when a small change made to a parameter value of a system causes a sudden qualitative or topological change in its behavior." (Dmitriy Laschov & Michael Margaliot, "Mathematical Modeling of the λ Switch: A Fuzzy Logic Approach", 2010)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory",  2011)

"Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behavior. Bifurcations can occur in both continuous systems (described by ODEs, DDEs, or PDEs) and discrete systems (described by maps)." (Tianshou Zhou, "Bifurcation", 2013)

"Roughly speaking, refers to the phenomenon of a system exhibiting new dynamical behavior as the parameter is varied." (Wei-Bin Zhang, "Chaos in Economics", 2014)

"A sudden change that accompanies the onset of chaos at a critical value of a varied control parameter." (Viet-Thanh Pham et al, "Chaotic Attractor in a Novel Time-Delayed System with a Saturation Function", 2015) 

"A qualitative change in the behavior of a dynamic system." (Ben Tran, "Enneagram through Chaos Theory", 2016)