A Representational Approach to Reduction in Dynamical Systems (original) (raw)

Reduction in Dynamical Systems: A Representational View

2010

ABSTRACT: 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.

Emulation, Reduction, and Emergence in Dynamical Systems

The received view about emergence and reduction is that they are incompatible categories. I argue in this paper that, contrary to the received view, emergence and reduction can hold together. To support this thesis, I focus attention on dynamical systems and, on the basis of a general representation theorem, I argue that, as far as these systems are concerned, the emulation relationship is sufficient for reduction (intuitively, a dynamical system DS1 emulates a second dynamical system DS2 when DS1 exactly reproduces the whole dynamics of DS2). This representational view of reduction, contrary to the standard deductivist one, is compatible with the existence of structural properties of the reduced system that are not also properties of the reducing one. Therefore, under this view, by no means are reduction and emergence incompatible categories but, rather, complementary ones.

Theory Reduction in Physics: A Model-Based, Dynamical Systems Approach

In 1973, Nickles identified two senses in which the term 'reduction' is used to describe the relationship between physical theories: namely, the sense based on Nagel's seminal account of reduction in the sciences, and the sense that seeks to extract one physical theory as a mathematical limit of another. These two approaches have since been the focus of most literature on the subject, as evidenced by recent work of Batterman and Butterfield, among others. In this paper, I discuss a third sense in which one physical theory may be said to reduce to another. This approach, which I call 'dynamical systems (DS) reduction,' concerns the reduction of individual models of physical theories rather than the wholesale reduction of entire theories, and specifically reduction between models that can be formulated as dynamical systems. DS reduction is based on the requirement that there exist a function from the state space of the low-level (more encompassing) model to that of the high-level (less encompassing) model that satisfies certain general constraints and thereby serves to identify quantities in the low-level model that mimic the behavior of those in the high-level model -but typically only when restricted to a certain domain of parameters and states within the low-level model. I discuss the relationship of this account of reduction to the Nagelian and limit-based accounts, arguing that it is distinct from both but exhibits strong parallels with a particular version of Nagelian reduction, and that the domain restrictions employed by the DS approach may, but need not, be specified in a manner characteristic of the limit-based approach. Finally, I consider some limitations of the account of reduction that I propose and suggest ways in which it might be generalised. I offer a simple, idealised example to illustrate application of this approach; a series of more realistic case studies of DS reduction is presented in another paper.

Approximate Reduction of Dynamical Systems

Proceedings of the 45th IEEE Conference on Decision and Control, 2006

The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction is typically performed in an "exact" manner-as is the case with mechanical systems with symmetry-which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples.

A GENERALIZED REDUCTION PROCEDURE FOR DYNAMICAL SYSTEMS

Modern Physics Letters A, 1991

A generalized reduction procedure for general dynamical systems is presented in an algebraic setting. This procedure extends previously defined reduction procedures for symplectic and Poisson systems. The algebraic formulation of this procedure allow to extend it to include Nambu structures as well and a reduction theorem for Nambu rings is proved.

Dynamical reduction models

Physics Reports, 2003

The report presents an exhaustive review of the recent attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and stochastic modifications of the Schrödinger equation. The proposed new dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wavepacket reduction and for the classical behavior of macroscopic systems. We recall the motivations for the new approach and we briefly review the other proposals to circumvent the above mentioned difficulties which appeared in the literature. In this way we make clear the conceptual and historical context characterizing the new approach. After having reviewed the mathematical techniques (stochastic differential calculus) which are essential for the rigorous and precise formulation of the new dynamics, we discuss in great detail its implications and we stress its relevant conceptual achievements. The new proposal requires also to work out an appropriate interpretation; a procedure which leads us to a reconsideration of many important issues about the conceptual status of theories based on a genuinely Hilbert space description of natural processes. Attention is also paid to many problems which are naturally raised by the dynamical reduction program. In particular we discuss the possibility and the problems one meets in trying to develop an analogous formalism for the relativistic case. Finally we discuss the experimental implications of the new dynamics for various physical processes which should allow, in principle, to test it against quantum mechanics. The review covers the work which has been done in the last fifteen years by various scientists and the lively debate which has accompanied the elaboration of the new proposal.

Approximate reduction of dynamic systems

Systems & Control Letters, 2008

The reduction of dynamical systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an "exact" manner-as is the case with mechanical systems with symmetry-which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamical system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e., when it is behaviorally similar to a dynamical system on a lower dimensional space. These concepts are illustrated on a series of examples. arXiv:0707.3804v1 [math.OC]

Approximate reduction of dynamic systemsI

The reduction of dynamic systems has a rich history, with many important applications related to stability, control and verification. Reduction of nonlinear systems is typically performed in an "exact" manner -as is the case with mechanical systems with symmetry -which, unfortunately, limits the type of systems to which it can be applied. The goal of this paper is to consider a more general form of reduction, termed approximate reduction, in order to extend the class of systems that can be reduced. Using notions related to incremental stability, we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors, i.e. when it is behaviourally similar to a dynamic system on a lower dimensional space. These concepts are illustrated on a series of examples.

Reduction as an A Posteriori Relation

Reduction between theories in physics is often approached as an a priori relation in the sense that reduction is often taken to depend only on a comparison of the mathematical structures of two theories. I argue that such purely formal approaches fail to capture one crucial sense of 'reduction,' whereby one theory encompasses the set of real behaviors that are well-modeled by the other. Reduction in this sense depends not only on the mathematical structures of the theories but also on empirical facts about where the theories succeed at describing real systems, and is therefore an a posteriori relation. I discuss several detailed implications of this claim for the methodology of inter-theory and inter-model reduction in physics.

Reductionist and anti-reductionist perspectives on dynamics

Philosophical Psychology, 2002

In this paper reduction and its pragmatics are discussed in the light of the development in Computer Science of languages to describe processes. The design of higher-level description languages within Computer Science has had the aim of allowing for description of the dynamics of processes in the (physical) world on a higher level avoiding all (physical) details of these processes. The higher description levels developed have dramatically increased the complexity of applications that came within reach. The pragmatic attitude of a (scientific) practitioner in this area has become inherently anti-reductionist, but based on well-established reduction relations. The paper discusses how this perspective can be related to reduction in general, and to other domains where description of dynamics plays a main role, in particular, biological and cognitive domains.