Dynamical Systems on Monoids: Toward a General Theory of Deterministic Systems and Motion

Giunti, Marco; Mazzola, Claudio

doi:10.1142/9789814383332_0012

Cited by 10 publications

(5 citation statements)

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“…An autonomous deterministic system is typically understood as a group of transformations on a given set, along with the postulate of their invariance with respect to time displacement (Lucas 1973 [4]; van Fraassen 1989 [9]). However, according to Giunti and Mazzola [2] (this volume), a dynamical system on a monoid is the minimal mathematical structure needed to capture the intuitive concept of a deterministic system. Giunti and Mazzola define a dynamical system on a monoid as follows:…”

Section: Dynamical Systems On Monoidsmentioning

confidence: 99%

“…From an intuitive point of view, for any t T and i I, the arrow i t →  t (i) (see sec. 3 of Giunti and Mazzola [2], this volume) is intended to represent the flowing of time from the present instant i to the instant  t (i) reached after duration t; therefore, on this interpretation,  t (i) cannot be anything else but the instant obtained by adding duration t to instant i, just as required by (ii) of Definition 4. It is easy to prove that the time system of any monoid is a dynamical system on the monoid itself.…”

Section: Time Models and Time Systemsmentioning

confidence: 99%

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Reversible Dynamics and the Directionality of Time

Mazzola¹,

Giunti²

2012

Methods, Models, Simulations and Approaches Towards a General Theory of Change

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The received view on the problem of the direction of time holds it that time has no intrinsic dynamical properties, and that its apparent asymmetry, to be understood in purely topological terms, is dependent on the directional properties of physical processes. In this paper we shall challenge both claims, in the light of an algebraic representation of time. First, we will show how to give a precise formulation to the intuitive idea that time possesses an intrinsic dynamics; this formulation relies on the fact that the algebraic properties of time can equivalently be understood in dynamical terms. Second, we shall argue that the directional properties displayed by the processes occurring in time depend on the directional properties of time, rather than the converse.

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Section: Dynamical Systems On Monoidsmentioning

confidence: 99%

Section: Time Models and Time Systemsmentioning

confidence: 99%

Reversible Dynamics and the Directionality of Time

Mazzola¹,

Giunti²

2012

Methods, Models, Simulations and Approaches Towards a General Theory of Change

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“…3; then, according to the previous definitions, I He φ is an empirical interpretation of DS e on H eφ , and (DS e , I He φ ) = DS e φ is an empirical model of H eφ . If φ is sufficiently small, and for an appropriate value of the constant g , such a model also turns out to be empirically correct, 14 and it is thus an example of a Galilean model.…”

mentioning

confidence: 93%

“…3 Def. 1 can be made completely general by taking the time set T to be the domain of an arbitrary monoid L = (T, +) (Giunti and Mazzola 2010). For the purposes of this paper, however, it suffice to consider dynamical systems whose time set T is either Z, Z + , R, or R + .…”

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confidence: 99%

A Representational Approach to Reduction in Dynamical Systems

Giunti

2014

Erkenn

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According to the received view, reduction is a deductive relation between two formal theories. In this paper, I develop an alternative approach, according to which reduction is a representational relation between models, rather than a deductive relation between theories; more specifically, I maintain that this representational relation is the one of emulation. To support this thesis, I focus attention on mathematical dynamical systems and I argue that, as far as these systems are concerned, the emulation relation is sufficient for reduction. I then extend this representational view of reduction to the case of empirically interpreted dynamical systems, as well as to a treatment of partial, approximate, and asymptotic reduction.

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“…In this thesis, we consider dynamical systems with continuous time and follow the definition in [67][68][69], where a dynamical system is defined as a tuple (S, T, Φ):…”

Section: Dynamical Systemsmentioning

confidence: 99%

Dynamic Responses of Networks under Perturbations: Solutions, Patterns and Predictions

Zhang¹

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The myriad things rise and fall, I thereby observe and contemplate the Law.All things bloom and flourish, then return to their roots, each by each.Laozi "Dao De Jing" (400 BC) knowledge on the behavior of a single starling does not ensure the comprehension of their flocking behavior. Fascinating phenomena such as the weather, life, and the flocking of birds arise from large collections of simpler components, but are much more than the sum of them: the dynamical interaction between the parts gives rise to the complexity of the whole. media [34]. Novel classes of patterns emerge if the underlying network architecture is further generalized. Patterns such as travelling waves and quasi-stationary patterns can be induced in directed, hence non-symmetric networks [35] and distinct heterogeneous patterns occur in multiplex networks [36].Thus, we can discern that, despite the importance and ubiquity of the problem of perturbation spreading, general answers to simple questions like "When and how will a perturbation affect a given node?" are still missing to date. Even for networks of linear deterministic dynamical systems, the questions remain open [64]. Synopsis of the thesisIn this thesis, we develop a theory of dynamic response patterns for complex networks under the influence of fluctuating perturbations. Motivated by the functional robustness Part I. FundamentalsChapter 1 Networks as Dynamical SystemsAs the beginning of the thesis, we present the most fundamental concepts underlying the study of dynamic network responses in this thesis. We start with basic and relevant concepts in graph theory, such as graphs, paths and distances (Sec. 1.1). Particularly, we highlight the relation between the distance between a pair of nodes and the element of the power of Laplacian matrices associated to the node pair. In Sec. 1.2, we move on to another branch of mathematics, i.e. dynamical system theory. We present the definition of dynamical systems and focus on the linear stability analysis at fixed points. In the last section of the chapter, Sec. 1.3, we combine the aforementioned concepts from the two branches and present basic considerations about the dynamics of a networked system, i.e. a system of coupled dynamical systems. We especially focus on the linearization of the dynamics of a general network system with pair-wise coupling at the fixed point. NetworksNetworks are closely related to the concept graph in mathematical literature, which is defined as a collection of vertices connected by edges, representing how a set of objects are related to each other (Fig. 1.1). Vertices and edges are sometimes also called nodes and links, or sites and bonds [4]. The three terms "vertices", "nodes" and "sites" referring to the same notion are interchangeable in this thesis, as well as "edges", "links", and "bonds". The matrix representation of graphsMathematically, a graph G(V, E) consists of a non-empty finite set V of vertices and a finite family E of edges, which are unordered pairs of vertices [5]. In this thesis we consider only ...

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Dynamical Systems on Monoids: Toward a General Theory of Deterministic Systems and Motion

Cited by 10 publications

References 5 publications

Reversible Dynamics and the Directionality of Time

Reversible Dynamics and the Directionality of Time

A Representational Approach to Reduction in Dynamical Systems

Dynamic Responses of Networks under Perturbations: Solutions, Patterns and Predictions

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