scholarly journals Product dynamics for homoclinic attractors

2005 ◽  
Vol 461 (2053) ◽  
pp. 155-177 ◽  
Author(s):  
Peter Ashwin ◽  
Michael Field
Keyword(s):  
Analytic Description ◽  
Small Subset ◽  
Direct Products ◽  
Asymptotically Stable ◽  
Product System ◽  
Hyperbolic Attractor ◽  
Product Systems ◽  
Homoclinic Cycle ◽  
Structurally Stable ◽  
Generic Behaviour

Heteroclinic cycles may occur as structurally stable asymptotically stable attractors if there are invariant subspaces or symmetries of a dynamical system. Even for cycles between equilibria, it may be difficult to obtain results on the generic behaviour of trajectories converging to the cycle. For more–complicated cycles between chaotic sets, the non–trivial dynamics of the ‘nodes’ can interact with that of the ‘connections’. This paper focuses on some of the simplest problems for such dynamics where there are direct products of an attracting homoclinic cycle with various types of dynamics. Using a precise analytic description of a general planar homoclinic attractor, we are able to obtain a number of results for direct product systems. We show that for flows that are a product of a homoclinic attractor and a periodic orbit or a mixing hyperbolic attractor, the product of the attractors is a minimal Milnor attractor for the product. On the other hand, we present evidence to show that for the product of two homoclinic attractors, typically only a small subset of the product of the attractors is an attractor for the product system.

scholarly journals ON PRODUCT SYSTEMS ARISING FROM SUM SYSTEMS

2005 ◽  
Vol 08 (01) ◽  
pp. 1-31 ◽  
Author(s):  
B. V. RAJARAMA BHAT ◽  
R. SRINIVASAN
Keyword(s):  
Hilbert Spaces ◽  
Von Neumann Algebra ◽  
Type I ◽  
Product System ◽  
Von Neumann ◽  
Type Iii ◽  
Product Systems ◽  
Weyl Representation ◽  
Uncountable Family ◽  
Type Spaces

B. Tsirelson constructed an uncountable family of type III product systems of Hilbert spaces through the theory of Gaussian spaces, measure type spaces and "slightly colored noises", using techniques from probability theory. Here we take a purely functional analytic approach and try to have a better understanding of Tsireleson's construction and his examples. We prove an extension of Shale's theorem connecting symplectic group and Weyl representation. We show that the "Shale map" respects compositions (this settles an old conjecture of K. R. Parthasarathy8). Using this we associate a product system to a sum system. This construction includes the exponential product system of Arveson, as a trivial case, and the type III examples of Tsirelson. By associating a von Neumann algebra to every "elementary set" in [0, 1], in a much simpler and direct way, we arrive at the invariants of the product system introduced by Tsirelson, given in terms of the sum system. Then we introduce a notion of divisibility for a sum system, and prove that the examples of Tsirelson are divisible. It is shown that only type I and type III product systems arise out of divisible sum systems. Finally, we give a sufficient condition for a divisible sum system to give rise to a unitless (type III) product system.

scholarly journals Grazing-Sliding Bifurcations Creating Infinitely Many Attractors

2017 ◽  
Vol 27 (12) ◽  
pp. 1730042 ◽  
Author(s):  
David J. W. Simpson
Keyword(s):  
Periodic Orbit ◽  
Periodic Orbits ◽  
Piecewise Linear ◽  
Simple Type ◽  
Asymptotically Stable ◽  
Sliding Bifurcation ◽  
Continuous Maps ◽  
Stable Periodic ◽  
Smooth System ◽  
Structurally Stable

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations, structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.

scholarly journals Discrete product systems with twisted units

1995 ◽  
Vol 52 (2) ◽  
pp. 317-326 ◽  
Author(s):  
Marcelo Laca
Keyword(s):  
Dynamical System ◽  
Crossed Product ◽  
Type I ◽  
Crossed Products ◽  
Product System ◽  
C Algebra ◽  
Covariant Representations ◽  
Product Systems ◽  
Recent Theorem ◽  
I Factor

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

scholarly journals KMS STATES ON NICA–TOEPLITZ ALGEBRAS OF PRODUCT SYSTEMS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250123 ◽  
Author(s):  
JEONG HEE HONG ◽  
NADIA S. LARSEN ◽  
WOJCIECH SZYMAŃSKI
Keyword(s):  
Finite Type ◽  
Lattice Ordered Group ◽  
Toeplitz Algebra ◽  
Product System ◽  
Natural Numbers ◽  
Ordered Group ◽  
Kms States ◽  
The Core ◽  
Product Systems ◽  
Affine Semigroup

We investigate KMS states of Fowler's Nica–Toeplitz algebra [Formula: see text] associated to a compactly aligned product system X over a semigroup P of Hilbert bimodules. This analysis relies on restrictions of these states to the core algebra which satisfy appropriate scaling conditions. The concept of product system of finite type is introduced. If (G, P) is a lattice ordered group and X is a product system of finite type over P satisfying certain coherence properties, we construct KMSβ states of [Formula: see text] associated to a scalar dynamics from traces on the coefficient algebra of the product system. Our results were motivated by, and generalize some of the results of Laca and Raeburn obtained for the Toeplitz algebra of the affine semigroup over the natural numbers.

scholarly journals Boundary value problems associated with first order rectangular: Kronecker product systems

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 45-53
Author(s):  
M.S.N. Murty ◽  
G. Srinivasu ◽  
Suresh Kumar
Keyword(s):  
Kronecker Product ◽  
General Boundary Condition ◽  
Product System ◽  
Existence And Uniqueness Results ◽  
Product Systems ◽  
Variation Of Parameters ◽  
Uniqueness Results ◽  
System P ◽  
Non Linear ◽  
Variation Of Parameters Formula

In this paper first, we establish a general solution of the non-linear Kronecker product system (P(Q)(t)y'(t)+(R(S)(t)y(t) = f(t, y(t)) with the help of variation of parameters formula. Finally, we prove existence and uniqueness results for the non-linear Kronecker product system satisfying general boundary condition Uy = ?, by using Schauder-Tychonov's and Brouwer's fixed point theorems.

Data-Driven Platform Design: Patent Data and Function Network Analysis

2018 ◽  
Vol 141 (2) ◽  
Author(s):  
Binyang Song ◽  
Jianxi Luo ◽  
Kristin Wood
Keyword(s):  
Network Analysis ◽  
Product Variety ◽  
Data Driven ◽  
Patent Data ◽  
Product System ◽  
Platform Design ◽  
Economy Of Scale ◽  
Product Systems ◽  
And Function ◽  
Platform System

A properly designed product-system platform seeks to reduce the cost and lead time for design and development of the product-system family. A key goal is to achieve a tradeoff between economy of scope from product variety and economy of scale from platform sharing. Traditionally, product platform planning uses heuristic and manual approaches and relies almost solely on expertise and intuition. In this paper, we propose a data-driven method to draw the boundary of a platform-system, complementing the other platform design approaches and assisting designers in the architecting process. The method generates a network of functions through relationships of their co-occurrences in prior designs of a product or systems domain and uses a network analysis algorithm to identify an optimal core–periphery structure. Functions identified in the network core co-occur cohesively and frequently with one another in prior designs, and thus, are suggested for inclusion in the potential platform to be shared across a variety of product-systems with peripheral functions. We apply the method to identify the platform functions for the application domain of spherical rolling robots (SRRs), based on patent data.

scholarly journals THE INDEX OF (WHITE) NOISES AND THEIR PRODUCT SYSTEMS

2006 ◽  
Vol 09 (04) ◽  
pp. 617-655 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hilbert Modules ◽  
Product System ◽  
Invariant Vector ◽  
Product Systems ◽  
System A ◽  
Definition Of ◽  
General Product ◽  
White Noises

Almost every paper about Arveson systems (i.e. product systems of Hilbert spaces) starts by recalling their basic classification assigning to every Arveson system a type and an index. So it is natural to ask in how far an analogue classification can also be proposed for product systems of Hilbert modules. However, while the definition of type is plain, there are obstacles for the definition of index. But all obstacles can be removed when restricting to the category which we introduce here as spatial product systems and that matches the usual definition of spatial in the case of Arveson systems. This is not really a loss because the definition of index for nonspatial Arveson systems is rather formal and does not reflect the information the index carries for spatial Arveson systems.E0-semigroups give rise to product systems. Our definition of spatial product system, namely, existence of a unital unit that is central, matches Powers' definition of spatial in the sense that the E0-semigroup from which the product system is derived admits a semigroup of intertwining isometries. We show that every spatial product system contains a unique maximal completely spatial subsystem (generated by all units) that is isomorphic to a product system of time ordered Fock modules. (There exist nonspatial product systems that are generated by their units. Consequently, these cannot be Fock modules.) The index of a spatial product system we define as the (unique) Hilbert bimodule that determines the Fock module. In order to show that the index merits the name index we provide a product of product systems under which the index is additive (direct sum). While for Arveson systems there is the tensor product, for general product systems the tensor product does not make sense as a product system. Even for Arveson systems our product is, in general, only a subsystem of the tensor product. Moreover, its construction depends explicitly on the choice of the central reference units of its factors.Spatiality of a product system means that it may be derived from an E0-semigroup with an invariant vector expectation, i.e. from a noise. We extend our product of spatial product systems to a product of noises and study its properties.Finally, we apply our techniques to show the module analogue of Fowler's result that free flows are comletely spatial, and we compute their indices.

Study on Industry Manufacturing with BN-Based Risk Assessment on the Technology Innovation of the Complex Product System

2013 ◽  
Vol 675 ◽  
pp. 8-12
Author(s):  
Nai Ping Song ◽  
Xian Bin Wu ◽  
Wei Cong Zhang ◽  
Ai Qin Huang
Keyword(s):  
Bayesian Network ◽  
Product Innovation ◽  
Rapid Development ◽  
Product System ◽  
Complex Product ◽  
Product Systems ◽  
Complex Product Systems ◽  
High Investment ◽  
Complex Technology ◽  
Research Period

With the rapid development of information technology and the accelerated pace of global economic integration, Industry Manufacturing of complex product systems (CoPS, Complex Product System) has become the “engine” for the country's technological progress and economic development. In order to ensure the success of industry product innovation, it is necessary to evaluate the risks in the process of innovation due to the high investment, complex technology, long research period, and high cost. This paper aims to explore the innovation risks by means of Bayesian network (BN, Bayesian Network) system.

ON THE EXISTENCE OF E0-SEMIGROUPS

2006 ◽  
Vol 09 (02) ◽  
pp. 315-320 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Direct Proof ◽  
Product System ◽  
Indirect Methods ◽  
Product Systems ◽  
Elementary Construction

Product systems are the classifying structures for semigroups of endomorphisms of [Formula: see text], in that two E0-semigroups are cocycle conjugate iff their product systems are isomorphic. Thus it is important to know that every abstract product system is associated with an E0-semigroup. This was first proved more than 15 years ago by rather indirect methods. Recently, Skeide has given a more direct proof. In this note we give yet another proof by an elementary construction.

A Connes correspondence approach to the dilation theory

2020 ◽  
Vol 31 (05) ◽  
pp. 2050040
Author(s):  
Yusuke Sawada
Keyword(s):  
Von Neumann Algebra ◽  
Classification Theory ◽  
Type I ◽  
View Point ◽  
Product System ◽  
Von Neumann ◽  
Product Systems ◽  
Dilation Theory ◽  
One To One

Product systems have been originally introduced to classify E0-semigroups on type I factors by Arveson. The idea of product systems is influenced the constructions of dilations of CP0-semigroups. In this paper, we will develop the dilation theory and the classification theory of E0-semigroups on a general von Neumann algebra in the view point of Connes correspondences. For this, we will provide a concept of product system whose components are Connes correspondences. There exists a one-to-one correspondence between a CP0-semigroup and a unit of a product system of Connes correspondences. The correspondence enables us to construct dilations and classify E0-semigroups by product systems of Connes correspondences up to cocycle equivalence.

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