On Bifurcations III

"At the large scale where many processes and structures appear continuous and stable much of the time, important changes may occur discontinuously. When only a few variables are involved, as well as an optimizing process, the event may be analyzed using catastrophe theory. As the number of variables in- creases the bifurcations can become more complex to the point where chaos theory becomes the relevant approach. That chaos theory as well as the fundamentally discontinuous quantum processes may be viewed through fractal eyeglasses can also be admitted. We can even argue that a cascade of bifurcations to chaos contains two essentially structural catastrophe points, namely the initial bifurcation point at which the cascade commences and the accumulation point at which the transition to chaos is finally achieved." (J Barkley Rosser Jr., "From Catastrophe to Chaos: A General Theory of Economic Discontinuities", 1991)

"The idea of one description of a system bifurcating from another also provides the key to begin unlocking one of the most important, and at the same time perplexing, problems of system theory: characterization of the complexity of a system." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"The key to making discontinuity emerge from smoothness is the observation that the overall behavior of both static and dynamical systems is governed by what's happening near the critical points. These are the points at which the gradient of the function vanishes. Away from the critical points, the Implicit Function Theorem tells us that the behavior is boring and predictable, linear, in fact. So it's only at the critical points that the system has the possibility of breaking out of this mold to enter a new mode of operation. It's at the critical points that we have the opportunity to effect dramatic shifts in the system's behavior by 'nudging' lightly the system dynamics, one type of nudge leading to a limit cycle, another to a stable equilibrium, and yet a third type resulting in the system's moving into the domain of a 'strange attractor'. It's by these nudges in the equations of motion that the germ of the idea of discontinuity from smoothness blossoms forth into the modern theory of singularities, catastrophes and bifurcations, wherein we see how to make discontinuous outputs emerge from smooth inputs." (John L Casti, "Reality Rules: Picturing the world in mathematics", 1992)

"Whenever patterns are perceived in a process, there is the possibility of extrapolation. Whatever the nature of the pattern, it provides a handle for grasping something about the way it will unfold in the future." (Ervin László, "Vision 2020: Reordering Chaos for Global Survival", 1994)

"In many nonlinear systems, however, small changes of certain parameters may produce Dramatic changes in the basic characteristics of the phase portrait. Attractors may disappear, or change into one another, or new attractors may suddenly appear. Such systems are said to be structurally unstable, and the critical points of instability are called 'bifurcation points', because they are points in the system’s evolution where a fork suddenly appears and the system branches off in a new direction. Mathematically, bifurcation points mark sudden changes in the system’s phase portrait. Physically, they correspond to points of instability at which the system changes abruptly and new forms of order suddenly appear." (Fritjof Capra, "The Systems View of Life: A Unifying Vision", 2014)

"Bifurcation is a qualitative, topological change of a system’s phase space that occurs when some parameters are slightly varied across their critical thresholds. Bifurcations play important roles in many real-world systems as a switching mechanism. […] There are two categories of bifurcations. One is called a local bifurcation, which can be characterized by a change in the stability of equilibrium points. It is called local because it can be detected and analyzed only by using localized information around the equilibrium point. The other category is called a global bifurcation, which occurs when non-local features of the phase space, such as limit cycles (to be discussed later), collide with equilibrium points in a phase space. This type of bifurcation can’t be characterized just by using localized information around the equilibrium point."  (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

Out of Context: On Bifurcation (Definitions)

"[…] bifurcations - the abrupt changes that can take place in the behavior, and often in the complexity, of a system when the value of a constant is altered slightly." (Edward N Lorenz, "The Essence of Chaos", 1993)

"A bifurcation is an event that occurs in the evolution of a dynamic system in which the characteristic behavior of the system is transformed." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"The concept of bifurcation, present in the context of non-linear dynamic systems and theory of chaos, refers to the transition between two dynamic modalities qualitatively distinct; both of them are exhibited by the same dynamic system, and the transition (bifurcation) is promoted by the change in value of a relevant numeric parameter of such system." (Emilio Del-Moral-Hernandez, "Chaotic Neural Networks", Encyclopedia of Artificial Intelligence, 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)

"The qualitative structure of the flow can change as parameters are varied. In particular, fixed points can be created or destroyed, or their stability can change. These qualitative changes in the dynamics are called bifurcations, and the parameter values at which they occur are called bifurcation points." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)

"[…] what exactly do we mean by a bifurcation? The usual definition involves the concept of 'topological equivalence': if the phase portrait changes its topological structure as a parameter is varied, we say that a bifurcation has occurred." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)