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The authors consider almost periodic differential equations and establish conditions for their discrete counterpart such that, if the discrete solution is almost periodic, then its continuous counterpart will be almost periodic. They relax the condition on the observation set, i.e. the set does not need to possess the structure of a subgroup, which was assumed by many authors earlier. At the end, an application to a time varying system is given.
Introduction to dynamical systems
2007
One approach starts from time-continuous differential equations and leads to time-discrete maps, which are obtained from them by a suitable discretization of time. This path is pursued, eg, in the book by Strogatz [Str94]. 1 The other approach starts from the study of time-discrete maps and then gradually builds up to time-continuous differential equations, see, eg,[Ott93, All97, Dev89, Has03, Rob95].
Hidden Attractors in Discrete Dynamical Systems
Entropy
Research using chaos theory allows for a better understanding of many phenomena modeled by means of dynamical systems. The appearance of chaos in a given process can lead to very negative effects, e.g., in the construction of bridges or in systems based on chemical reactors. This problem is important, especially when in a given dynamic process there are so-called hidden attractors. In the scientific literature, we can find many works that deal with this issue from both the theoretical and practical points of view. The vast majority of these works concern multidimensional continuous systems. Our work shows these attractors in discrete systems. They can occur in Newton’s recursion and in numerical integration.
Qualitative Methods in Continuous and Discrete Dynamical Systems
Springer Proceedings in Complexity, 2016
This volume contains the lessons delivered during the "Training School on qualitative theory of dynamical systems, tools and applications" held at the University of Urbino (Italy) from 17 September to 19 September 2015 in the framework of the European COST Action "The EU in the new complex geography of economic systems: models, tools and policy evaluation" (Gecomplexity). Gecomplexity is a European research network, inspired by the New Economic Geography approach, initiated by P. Krugman in the early 1990s, which describes economic systems as multilayered and interconnected spatial structures. At each layer, different types of decisions and interactions are considered: interactions between international or regional trading partners at the macrolevel; the functioning of (financial, labour, goods) markets as social network structures at mesolevel; and finally, the location choices of single firms at the microlevel. Within these structures, spatial inequalities are evolving through time following complex patterns determined by economic, geographical, institutional and social factors. In order to study these structures, the Action aims to build an interdisciplinary approach to develop advanced mathematical and computational methods and tools for analysing complex nonlinear systems, ranging from social networks to game theoretical models, with the formalism of the qualitative theory of dynamical systems and the related concepts of attractors, stability, basins of attraction, local and global bifurcations. Following the same spirit, this book should provide an introduction to the study of dynamic models in economics and social sciences, both in discrete and in continuous time, by the methods of the qualitative theory of dynamical systems. At the same time, the students should also practice (and, hopefully, appreciate) the interdisciplinary "art of mathematical modelling" of real-world systems and time-evolving processes. Indeed, the setup of a dynamic model of a real evolving system (physical, biological, social, economic, etc.) starts from a rigorous and critical analysis of the system, its main features and basic principles. Measurable quantities (i.e. quantities that can be expressed by numbers) that characterize its state and its behaviour must be identified in order to describe the system v Preface vii
Differential Equations, Dynamical Systems, and an Introduction to Chaos, 2013
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright
Representation in Dynamical Systems
arXiv (Cornell University), 2021
The brain is often called a computer and likened to a Turing machine, in part because the mind can manipulate discrete symbols such as numbers. But the brain is a dynamical system, more like a Watt governor than a Turing machine. Can a dynamical system be said to operate using "representations"? This paper argues that it can, although not in the way a digital computer does. Instead, it uses phenomena best described using mathematic concepts such as chaotic attractors to stand in for aspects of the world.
Partially-Observed Discrete Dynamical Systems
American Control Conference (ACC), 2021
This paper introduces a new signal model called partially-observed discrete dynamical systems (PODDS). This signal model is a special case of the hidden Markov model (HMM), where the state is a vector containing the information of different components of the system, and each component takes its value from a finite real-valued set. This signal model is currently treated as a finite-state HMM, where maximum a posteriori (MAP) criterion is used for state estimator purpose. This paper takes advantage of the discrete structure of the state variables in PODDS and develops the optimal component-wise MAP (CMAP) state estimator, which yields the MAP solution in each state variable. A fully-recursive process is provided for computation of this optimal estimator, followed by introducing a specific instance of the PODDS model suitable for regulatory networks observed through noisy time series data. The high performance of the proposed estimator is demonstrated by numerical experiments with a PODDS model of random regulatory networks.
Models of knowing and the investigation of dynamical systems
Physica D: Nonlinear Phenomena, 1999
We present three distinct concepts of what constitutes a scientific understanding of a dynamical system. The development of each of these paradigms has resulted in a significant expansion in the kind of system that can be investigated. In particular, the recently-developed 'algorithmic modelling paradigm' has allowed us to enlarge the domain of discourse to include complex real-world processes that cannot necessarily be described by conventional differential equations.
Unconventional Approaches, 2005
Classical dynamics concepts are analysed in the basic mathematical setting of state transition systems where time and space are both completely discrete and no structure is assumed on the state's space. Interesting relationships between attractors and recurrence are identified and some features of chaos are expressed in simple, set theoretic terms. String dynamics is proposed as a unifying concept for dynamical systems arising from discrete models of computation, together with illustrative examples. The relevance of state transition systems and string dynamics is discussed from the perspective of molecular computing.