On Phase Spaces

 "Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space. For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions. (Edward N Lorenz, "Deterministic Nonperiodic Flow", Journal of the Atmospheric Science 20, 1963)

"The ‘eyes of the mind’ must be able to see in the phase space of mechanics, in the space of elementary events of probability theory, in the curved four-dimensional space-time of general relativity, in the complex infinite dimensional projective space of quantum theory. To comprehend what is visible to the ‘actual eyes’, we must understand that it is only the projection of an infinite dimensional world on the retina." (Yuri I Manin, "Mathematics and Physics", 1981)

"[…] physicists have come to appreciate a fourth kind of temporal behavior: deterministic chaos, which is aperiodic, just like random noise, but distinct from the latter because it is the result of deterministic equations. In dynamic systems such chaos is often characterized by small fractal dimensions because a chaotic process in phase space typically fills only a small part of the entire, energetically available space." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990)

"When a system has more than one attractor, the points in phase space that are attracted to a particular attractor form the basin of attraction for that attractor. Each basin contains its attractor, but consists mostly of points that represent transient states. Two contiguous basins of attraction will be separated by a basin boundary." (Edward N Lorenz, "The Essence of Chaos", 1993)

"Chaos appears in both dissipative and conservative systems, but there is a difference in its structure in the two types of systems. Conservative systems have no attractors. Initial conditions can give rise to periodic, quasiperiodic, or chaotic motion, but the chaotic motion, unlike that associated with dissipative systems, is not self-similar. In other words, if you magnify it, it does not give smaller copies of itself. A system that does exhibit self-similarity is called fractal. [...] The chaotic orbits in conservative systems are not fractal; they visit all regions of certain small sections of the phase space, and completely avoid other regions. If you magnify a region of the space, it is not self-similar." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"One of the reasons we deal with the pendulum is that it is easy to plot its motion in phase space. If the amplitude is small, it's a two-dimensional problem, so all we need to specify it completely is its position and its velocity. We can make a two-dimensional plot with one axis (the horizontal), position, and the other (the vertical), velocity." (Barry R Parker, "Chaos in the Cosmos: The stunning complexity of the universe", 1996)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing - seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 1997)

"The chance events due to deterministic chaos, on the other hand, occur even within a closed system determined by immutable laws. Our most cherished examples of chance - dice, roulette, coin-tossing – seem closer to chaos than to the whims of outside events. So, in this revised sense, dice are a good metaphor for chance after all. It's just that we've refined our concept of randomness. Indeed, the deterministic but possibly chaotic stripes of phase space may be the true source of probability." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)

"Roughly spoken, bifurcation theory describes the way in which dynamical system changes due to a small perturbation of the system-parameters. A qualitative change in the phase space of the dynamical system occurs at a bifurcation point, that means that the system is structural unstable against a small perturbation in the parameter space and the dynamic structure of the system has changed due to this slight variation in the parameter space." (Holger I Meinhardt, "Cooperative Decision Making in Common Pool Situations", 2012)

"The impossibility of predicting which point in phase space the trajectory of the Lorenz attractor will pass through at a certain time, even though the system is governed by deterministic equations, is a common feature of all chaotic systems. However, this does not mean that chaos theory is not capable of any predictions. We can still make very accurate predictions, but they concern the qualitative features of the system’s behavior rather than the precise values of its variables at a particular time. The new mathematics thus represents the shift from quantity to quality that is characteristic of systems thinking in general. Whereas conventional mathematics deals with quantities and formulas, nonlinear dynamics deals with qualities and patterns." (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)

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)

Hiroki Sayama - Collected Quotes

"A giant component is a connected component whose size is on the same order of magnitude as the size of the whole network. Network percolation is the appearance of such a giant component in a random graph, which occurs when the average node degree is above 1." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A good model is simple, valid, and robust. Simplicity of a model is really the key essence of what modeling is all about. The main reason why we want to build a model is that we want to have a shorter, simpler description of reality. [...] Validity of a model is how closely the model’s prediction agrees with the observed reality. This is of utmost importance from a practical viewpoint. If your model’s prediction doesn’t reasonably match the observation, the model is not representing reality and is probably useless. [...] finally, robustness of a model is how insensitive the model’s prediction is to minor variations of model assumptions and/or parameter settings. This is important because there are always errors when we create assumptions about, or measure parameter values from, the real world. If the prediction made by your model is sensitive to their minor variations, then the conclusion derived from it is probably not reliable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"A model is a simplified representation of a system. It can be conceptual, verbal, diagrammatic, physical, or formal (mathematical)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"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)

"Chaos can be understood as a dynamical process in which microscopic information hidden in the details of a system’s state is dug out and expanded to a macroscopically visible scale (stretching), while the macroscopic information visible in the current system’s state is continuously discarded (folding)." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Chaos is a long-term behavior of a nonlinear dynamical system that never falls in any static or periodic trajectories. [It] looks like a random fluctuation, but still occurs in completely deterministic, simple dynamical systems. [It] exhibits sensitivity to initial conditions. [It] occurs when the period of the trajectory of the system’s state diverges to infinity. [It] occurs when no periodic trajectories are stable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Complex systems are networks made of a number of components that interact with each other, typically in a nonlinear fashion. Complex systems may arise and evolve through self-organization, such that they are neither completely regular nor completely random, permitting the development of emergent behavior at macroscopic scales." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Emergence is a nontrivial relationship between the properties of a system at microscopic and macroscopic scales. Macroscopic properties are called emergent when it is hard to explain them simply from microscopic properties." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Equilibrium points are important for both theoretical and practical reasons. Theoretically, they are key points in the system’s phase space, which serve as meaningful references when we understand the structure of the phase space. And practically, there are many situations where we want to sustain the system at a certain state that is desirable for us. In such cases, it is quite important to know whether the desired state is an equilibrium point, and if it is, whether it is stable or unstable." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Mean-field approximation is a technique that ignores spatial relationships among components. It works quite well for systems whose parts are fully connected or randomly interacting with each other. It doesn’t work if the interactions are local or non-homogeneous, and/or if the system has a non-uniform pattern of states. In such cases, you could still use mean-field approximation as a preliminary, “zeroth-order” approximation, but you should not derive a final conclusion from it." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"[...] nonlinearity means that the outputs of a system are not given by a linear combination of the inputs. In the context of system behavior, the inputs and outputs can be the current and next states of the system, and if their relationship is not linear, the system is called a nonlinear system." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"One of the unique features of typical CA [ cellular automata] models is that time, space, and states of cells are all discrete. Because of such discreteness, the number of all possible state-transition functions is finite, i.e., there are only a finite number of “universes” possible in a given CA setting. Moreover, if the space is finite, all possible configurations of the entire system are also enumerable. This means that, for reasonably small CA settings, one can conduct an exhaustive search of the entire rule space or phase space to study the properties of all the 'parallel universes'." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Self-organization is a dynamical process by which a system spontaneously forms nontrivial macroscopic structures and/or behaviors over time." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The challenge in developing a model becomes particularly tough when it comes to the modeling of complex systems, because their unique properties (networks, nonlinearity, emergence, self-organization, etc.) are not what we are familiar with. We usually think about things on a single scale in a step-by-step, linear chain of reasoning, in which causes and effects are clearly distinguished and discussed sequentially. But this approach is not suitable for understanding complex systems where a massive amount of components are interacting with each other interdependently to generate patterns over a broad range of scales. Therefore, the behavior of complex systems often appears to contradict our everyday experiences." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"The sensitivity of chaotic systems to initial conditions is particularly well known under the moniker of the 'butterfly effect', which is a metaphorical illustration of the chaotic nature of the weather system in which 'a flap of a butterfly’s wings in Brazil could set off a tornado in Texas'. The meaning of this expression is that, in a chaotic system, a small perturbation could eventually cause very large-scale difference in the long run." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"There are several reasons why reaction-diffusion systems have been a popular choice among mathematical modelers of spatio-temporal phenomena. First, their clear separation between non-spatial and spatial dynamics makes the modeling and simulation tasks really easy. Second, limiting the spatial movement to only diffusion makes it quite straightforward to expand any existing non-spatial dynamical models into spatially distributed ones. Third, the particular structure of reaction-diffusion equations provides aneasy shortcut in the stability analysis (to be discussed in the next chapter). And finally, despite the simplicity of their mathematical form, reaction-diffusion systems can show strikingly rich, complex spatio-temporal dynamics. Because of these properties, reaction-diffusion systems have been used extensively for modeling self-organization of spatial patterns." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Trajectories of a deterministic dynamical system will never branch off in its phase space (though they could merge), because if they did, that would mean that multiple future states were possible, which would violate the deterministic nature of the system. No branching means that, once you specify an initial state of the system, the trajectory that follows is uniquely determined too. You can visually inspect where the trajectories are going in the phase space visualization. They may diverge to infinity, converge to a certain point, or remain dynamically changing yet stay in a confined region in the phase space from which no outgoing trajectories are running out. Such a converging point or a region is called an attractor." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)

"Variable rescaling is a technique to eliminate parameters from your model without losing generality. The basic idea is this: Variables that appear in your model represent quantities that are measured in some kind of units, but those units can be arbitrarily chosen without changing the dynamics of the system being modeled. This must be true for all scientific quantities that have physical dimensions - switching from inches to centimeters shouldn’t cause any change in how physics works! This means that you have the freedom to choose any convenient unit for each variable, some of which may simplify your model equations." (Hiroki Sayama, "Introduction to the Modeling and Analysis of Complex Systems", 2015)