Periodic orbit theory is an effective tool for the analysis of classical and quantum chaotic systems. In this paper we extend this approach to stochastic systems, in particular to mappings with additive noise. The theory is cast in the standard field theoretic formalism, and weak noise perturbation theory written in terms of Feynman diagrams. The result is a stochastic analog of the next-to-leading ¯h corrections to the Gutzwiller trace formula, with long time averages calculated from periodic orbits of the deterministic system. The perturbative corrections are computed analytically and tested numerically on a simple 1-dimensional system. 1

Hyperbolic Sets for Noninvertible Maps and Relations by Evelyn Sander This thesis presents a theory of hyperbolic structures and dynamics of smooth noninvertible maps and relations. In this context, it includes a new proof of the stable manifold theorem for fixed points, the shadowing lemma, and a version of the stable manifold theorem for hyperbolic sets. It also gives a description of some of the behavior of transverse homoclinic orbits for noninvertible maps and relations. ii Contents 1 Introduction 1 1.1 Preliminary remarks : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.2 Previous theory and applications : : : : : : : : : : : : : : : : : : : 2 1.2.1 The noninvertible stable manifold theorem : : : : : : : : : : 3 1.3 Hyperbolic sets, stable manifolds, and the shadowing lemma : : : : 4 1.4 Transverse homoclinic orbits : : : : : : : : : : : : : : : : : : : : : : 5 2 Applications 7 2.1 Difference methods : : : : : : : : : : : : : : : : : : : : : : : : : : : 7 2.2 Adaptive ...

the breakup of an invariant circle

We discuss the parametrized dynamics of two coupled recurrent neural networks comprising either additive sigmoid neurons in discrete time or biologically more plausible time-continuous leaky-integrate-and- re cells. General conditions for the existence of synchronized activity in such networks are given, which guarantee that corresponding neurons in both coupled sub-networks evolve synchronously. It is, in particular, demonstrated that even the coupling of totally di erent network structures can result in complex dynamics constrained to a synchronization manifold M. For additive sigmoid neurons the synchronized dynamics can be periodic, quasiperiodic as well as chaotic, and its stability can be determined by Lyapunov exponent techniques. For leaky-integrate-and- re cells synchronized orbits are typically periodic, often with an extremely long period duration. In addition to synchronized attractors there often co-exist asynchronous periodic, quasiperiodic and even chaotic attractors.

Abstract The Lotka-Volterra predator-prey model exemplifies the implicit and explicit assumptions managers often have regarding species interaction—populations are stable or fluctuate periodically. The reality is often much more complicated and there is overwhelming evidence that many populations fluctuate in a nonperiodic way. Using a discrete predator-prey model that generates chaos, it is possible to qualitatively mimic the interaction of some predator-prey populations. The implications of the paper are that managers should place greater emphasis on theoretical modeling and simulations, try to understand ecosystems and broad relationships between species rather than obtain minute details and data on individual populations, make management as flexible as possible to help people adjust to rapid changes in populations, employ mixed strategies so as to give options whatever the underlying dynamics, and, where appropriate, experiment with different strategies for different subpopulations to learn more about the effectiveness of alternative management approaches. Key words Chaos, management strategies, predator-prey models, renewable resources. If, along the road, you chance upon a bird’s nest, in any tree or on the ground, with fledglings or eggs and the mother sitting over the fledglings or on the eggs, do not take the mother together with her young. Let the mother go, and take only the young. Deuteronomy 22:6–7

Abstract. In this work we consider two models of two dimensional discrete systems subjected to three different types of coupling and analyse systematically the performance of each in realising synchronised states. We find that linear coupling effectively introduce control of chaos along with synchronisation, while synchronised chaotic states are possible with an additive parametric coupling scheme both being equally relevent for specific applications. The basin leading to synchronisation in the initial value plane and the choice of parameter values for synchronisation in the parameter plane are isolated in each case. PACS numbers: 05.45.Gg; 0.5.45.Xt Keywords: synchronisation, stability, Gumowski-Mira map, additive and parametric coupling Synchronisation in two dimensional systems 2 1.

Cataloguing-in-Publication entry

In this paper it is shown how to model times series by using random iterated neural networks with placedependent probabilities. The model assumes that the time series comes from a dynamical system which has a compact global attractor and a physical probability measure supported on the attractor. Also, an evolutionary algorithm is used to train a random iterated neural network that models a financial time series. INTRODUCTION In this paper, we present an extension of random iterated neural networks [14] that consider place-dependent probabilities. These networks are discrete random dynamical systems [2] defined by iterated random function systems [7] with place-dependent probabilities [5]. We show that these neural networks can be used to model time series that come from an underlying dynamics embedded on a space X ae R m , which has a compact global attractor and a physical measure supported on the attractor. The fact that the underlying dynamics for the time series has a global ...

the date of receipt and acceptance should be inserted later Abstract. We discuss the possibility of simultaneous and sequential synchronisation in vertical and horizontal arrays of unidirectionally coupled discrete systems. This is realized for the specific case of two dimensional Gumowski-Mira maps. The synchronised state can be periodic, thereby bringing in control of chaos, or chaotic for carefully chosen parameters of the participating units. The synchronised chaotic state is further characterised using variation of the time of synchronisation with coupling coefficient, size of the array etc. In the case of the horizontal array, the total time of synchronisation can be controlled by increasing the coupling coefficient step wise in small bunch of units. PACS. 05.45.Xt Synchronization; coupled oscillators – 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 1

Studying discrete dynamical systems