Complex systems present problems both in mathematical modelling and philosophical foundations. The study of complex systems represents a new approach to science that investigates how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment. The equations from which models of complex systems are developed generally derive from statistical physics, information theory and non-linear dynamics, and represent organized but unpredictable behaviors of natural systems that are considered fundamentally complex.
In this paper, we propose, discuss, and illustrate a computationally feasible definition of chaos which can be applied very generally to situations that are commonly encountered, including attractors, repellers, and non-periodically forced systems. This definition is based on an entropy-like quantity, which we call “expansion entropy,” and we define chaos as occurring when this quantity is positive. We relate and compare expansion entropy to the well-known concept of topological entropy to which it is equivalent under appropriate conditions. We also present example illustrations, discuss computational implementations, and point out issues arising from attempts at giving definitions of chaos that are not entropy-based.
Ecosystems exhibit surprising regularities in structure and function across terrestrial and aquatic biomes worldwide. We assembled a global data set for 2260 communities of large mammals, invertebrates, plants, and plankton. We find that predator and prey biomass follow a general scaling law with exponents consistently near ¾. This pervasive pattern implies that the structure of the biomass pyramid becomes increasingly bottom-heavy at higher biomass. Similar exponents are obtained for community production-biomass relations, suggesting conserved links between ecosystem structure and function. These exponents are similar to many body mass allometries, and yet ecosystem scaling emerges independently from individual-level scaling, which is not fully understood. These patterns suggest a greater degree of ecosystem-level organization than previously recognized and a more predictive approach to ecological theory.
The predator-prey power law: Biomass scaling across terrestrial and aquatic biomes Ian A. Hatton, Kevin S. McCann, John M. Fryxell, T. Jonathan Davies, Matteo Smerlak, Anthony R. E. Sinclair, Michel Loreau
Author Summary Given a body and an environment, what is the brain complexity needed in order to generate a desired set of behaviors? The general understanding is that the physical properties of the body and the environment correlate with the required brain complexity. More precisely, it has been pointed that naturally evolved intelligent systems tend to exploit their embodiment constraints and that this allows them to express complex behaviors with relatively concise brains. Although this principle of parsimonious control has been formulated quite some time ago, only recently one has begun to develop the formalism that is required for making quantitative statements on the sufficient brain complexity given embodiment constraints. In this work we propose a precise mathematical approach that links the physical and behavioral constraints of an agent to the required controller complexity. As controller architecture we choose a well-known artificial neural network, the conditional restricted Boltzmann machine, and define its complexity as the number of hidden units. We conduct experiments with a virtual six-legged walking creature, which provide evidence for the accuracy of the theoretical predictions.
Human societies use complexity -- within their institutions and technologies -- to address their various problems, and they need high-quality energy to create and sustain this complexity. But now greater complexity is producing diminishing returns in wellbeing, while the energetic cost of key sources of energy is rising fast. Simultaneously, humankind's problems are becoming vastly harder, which requires societies to deliver yet more complexity and thus consume yet more energy. Resolving this paradox is the central challenge of the 21st century. Thomas Homer-Dixon holds the CIGI Chair of Global Systems at the Balsillie School of International Affairs in Waterloo, Canada, and is a Professor at the University of Waterloo.
Italian New Public Management (NPM) has been mainly characterized by a political orientation toward power decentralization to local governments and privatization of public companies. Nowadays, local utilities in Italy are often run by joint stock companies controlled by public agencies such as Regional and Municipal Administrations. Due to this transformation, these companies must comply with a set of diverse expectations coming from a wide range of stakeholders, related to their financial, competitive and social performance. Such fragmented governance increases the presence of “wicked” problems in the decision-making sphere of these entities. Given this multi-level governance structure, how do these agents influence public services performance? In recent years, coordination and inter-institutional joint action have been identified as possible approaches for dealing with governance fragmentation and wicked problems deriving from it. How can we adapt a performance management perspective in order to help us reform the system and so have a better collaboration between the stakeholders involved? In order to address and discuss these research questions, a case study will be developed. The case concerns AMAT, the local utility providing the public transportation service in the Municipality of Palermo (Italy). The result of this study is a dynamic model including a set of performance indicators that help us in understanding the impact of the governing structure on the system’s performance.
We discuss how understanding the nature of chaotic dynamics allows us to control these systems. A controlled chaotic system can then serve as a versatile pattern generator that can be used for a range of application. Specifically, we will discuss the application of controlled chaos to the design of novel computational paradigms. Thus, we present an illustrative research arc, starting with ideas of control, based on the general understanding of chaos, moving over to applications that influence the course of building better devices.
“Chaos is found in greatest abundance wherever order is being sought. It always defeats order, because it is better organized” Terry Pratchett A brief review is presented of some recent findings in the theory of chaotic dynamics. We also prove a statement that could be naturally considered as a dual one to the Poincaré theorem on recurrences. Numerical results demonstrate that some parts of the phase space of chaotic systems are more likely to be visited earlier than other parts. A new class of chaotic focusing billiards is discussed that clearly violates the main condition considered to be necessary for chaos in focusing billiards.
An introductory conclusion The Systems Approach: principles A few years ago I wrote a blog post about anti-planning as an alternative to the systems approach. Part of the post was devoted to a number of principles of deception-perception. Churchman discusses their importance in the concluding chapter of The Systems Approach (TSA), which…
The nexus concept aims at extending ‘integrated management thinking’, which has been applied with varying success in diverse disciplines and has become especially popular in water resources management. UNU-FLORES developed an interactive platform, the Nexus Tools Platform, for inter-model comparison of existing modeling tools related to Water-Soil-Waste Nexus providing detailed model information and advanced filtering based on real-time visualizations.
Introduction to the Modeling and Analysis of Complex Systems introduces students to mathematical/computational modeling and analysis developed in the emerging interdisciplinary field of Complex Systems Science. Complex systems are systems made of a large number of microscopic components interacting with each other in nontrivial ways. Many real-world systems can be understood as complex systems, where critically important information resides in the relationships between the parts and not necessarily within the parts themselves. This textbook offers an accessible yet technically-oriented introduction to the modeling and analysis of complex systems. The topics covered include: fundamentals of modeling, basics of dynamical systems, discrete-time models, continuous-time models, bifurcations, chaos, cellular automata, continuous field models, static networks, dynamic networks, and agent-based models. Most of these topics are discussed in two chapters, one focusing on computational modeling and the other on mathematical analysis. This unique approach provides a comprehensive view of related concepts and techniques, and allows readers and instructors to flexibly choose relevant materials based on their objectives and needs. Python sample codes are provided for each modeling example.
Historical knowledge is essential to practical applications of ecological economics. Systems of problem solving develop greater complexity and higher costs over long periods. In time such systems either require increasing energy subsidies or they collapse. Diminishing returns to complexity in problem solving limited the abilities of earlier societies to respond sustainably to challenges, and will shape contemporary responses to global change. To confront this dilemma we must understand both the role of energy in sustaining problem solving, and our historical position in systems of increasing complexity.
This paper describes a new concept of cellular automata (CA). XCA consists of a set of arcs (edges). These arcs correspond to cells in CA. At a definite time, the arcs are connected to a directed graph. With each next time step, the arcs are exchanging their neighbors (adjacent arcs) according to rules that are dependent on the status of the adjacent arcs. With the extended cellular automaton (XCA) an artificial world may be simulated starting with a Big Bang. XCA does not require a grid like CA do. However, it can create one, just as the real universe after the big bang generated its own space, which previously did not exist. Examples with different rules show how manifold the concept of XCA is. Like the game of life simulates birth, survival, and death, this game should simulate a system that starts from a singularity, and evolves to a complex space.
Random walk is a fundamental concept with applications ranging from quantum physics to econometrics. Remarkably, one specific model of random walks appears to be ubiquitous across many fields as a tool to analyze transport phenomena in which the dispersal process is faster than dictated by Brownian diffusion. The Lévy-walk model combines two key features, the ability to generate anomalously fast diffusion and a finite velocity of a random walker. Recent results in optics, Hamiltonian chaos, cold atom dynamics, biophysics, and behavioral science demonstrate that this particular type of random walk provides significant insight into complex transport phenomena. This review gives a self-consistent introduction to Lévy walks, surveys their existing applications, including latest advances, and outlines further perspectives.
Lévy walks V. Zaburdaev, S. Denisov, and J. Klafter Rev. Mod. Phys. 87, 483
A new offering from SFI’s online education resource,Complexity Explorer, gives complexity enthusiasts quantitative tools for distinguishing the "complex" aspects of a system from the merely "complicated."
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