Koopman Operator and its Approximations for Systems with Symmetries

Nonlinear dynamical systems with symmetries exhibit a rich variety of behaviors, including complex attractor-basin portraits and enhanced and suppressed bifurcations. Symmetry arguments provide a way to study these collective behaviors and to simplify their analysis. The Koopman operator is an infinite dimensional linear operator that fully captures a system's nonlinear dynamics through the linear evolution of functions of the state space. Importantly, in contrast with local linearization, it preserves a system's global nonlinear features. We demonstrate how the presence of symmetries affects the Koopman operator structure and its spectral properties. In fact, we show that symmetry considerations can also simplify finding the Koopman operator approximations using the extended and kernel dynamic mode decomposition methods (EDMD and kernel DMD). Specifically, representation theory allows us to demonstrate that an isotypic component basis induces block diagonal structure in operator approximations, revealing hidden organization. Practically, if the data is symmetric, the EDMD and kernel DMD methods can be modified to give more efficient computation of the Koopman operator approximation and its eigenvalues, eigenfunctions, and eigenmodes. Rounding out the development, we discuss the effect of measurement noise

A Basic Framework for the Cryptanalysis of Digital Chaos-Based Cryptography

Chaotic cryptography is based on the properties of chaos as source of entropy. Many different schemes have been proposed to take advantage of those properties and to design new strategies to encrypt information. However, the right and efficient use of chaos in the context of cryptography requires a thorough knowledge about the dynamics of the selected chaotic system. Indeed, if the final encryption system reveals enough information about the underlying chaotic system it could be possible for a cryptanalyst to get the key, part of the key or some information somehow equivalent to the key just analyzing those dynamical properties leaked by the cryptosystem. This paper shows what those dynamical properties are and how a cryptanalyst can use them to prove the inadequacy of an encryption system for the secure exchange of information. This study is performed through the introduction of a series of mathematical tools which should be the basic framework of cryptanalysis in the context of digital chaos-based cryptography.Comment: 6 pages, 5 figure

Chaotic versus stochastic behavior in active-dissipative nonlinear systems

We study the dynamical state of the one-dimensional noisy generalized Kuramoto-Sivashinsky (gKS) equation by making use of time-series techniques based on symbolic dynamics and complex networks. We focus on analyzing temporal signals of global measure in the spatiotemporal patterns as the dispersion parameter of the gKS equation and the strength of the noise are varied, observing that a rich variety of different regimes, from high-dimensional chaos to pure stochastic behavior, emerge. Permutation entropy, permutation spectrum, and network entropy allow us to fully classify the dynamical state exposed to additive noise

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic R\"ossler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.Comment: 12 pages, 13 figure

Incremental and Transitive Discrete Rotations

A discrete rotation algorithm can be apprehended as a parametric application $f\_\alpha$ from \ZZ[i] to \ZZ[i], whose resulting permutation ``looks like'' the map induced by an Euclidean rotation. For this kind of algorithm, to be incremental means to compute successively all the intermediate rotate d copies of an image for angles in-between 0 and a destination angle. The di scretized rotation consists in the composition of an Euclidean rotation with a discretization; the aim of this article is to describe an algorithm whic h computes incrementally a discretized rotation. The suggested method uses o nly integer arithmetic and does not compute any sine nor any cosine. More pr ecisely, its design relies on the analysis of the discretized rotation as a step function: the precise description of the discontinuities turns to be th e key ingredient that will make the resulting procedure optimally fast and e xact. A complete description of the incremental rotation process is provided, also this result may be useful in the specification of a consistent set of defin itions for discrete geometry

Spreading Sequences Generated Using Asymmetrical Integer-Number Maps

Chaotic sequences produced by piecewise linear maps can be transformed to binary sequences. The binary sequences are optimal for the asynchronous DS/CDMA systems in case of certain shapes of the maps. This paper is devoted to the one-to-one integer-number maps derived from the suitable asymmetrical piecewise linear maps. Such maps give periodic integer-number sequences, which can be transformed to the binary sequences. The binary sequences produced via proposed modified integer-number maps are perfectly balanced and embody good autocorrelation and crosscorrelation properties. The number of different binary sequences is sizable. The sequences are suitable as spreading sequences in DS/CDMA systems

Morse theory on spaces of braids and Lagrangian dynamics

In the first half of the paper we construct a Morse-type theory on certain spaces of braid diagrams. We define a topological invariant of closed positive braids which is correlated with the existence of invariant sets of parabolic flows defined on discretized braid spaces. Parabolic flows, a type of one-dimensional lattice dynamics, evolve singular braid diagrams in such a way as to decrease their topological complexity; algebraic lengths decrease monotonically. This topological invariant is derived from a Morse-Conley homotopy index and provides a gloablization of `lap number' techniques used in scalar parabolic PDEs. In the second half of the paper we apply this technology to second order Lagrangians via a discrete formulation of the variational problem. This culminates in a very general forcing theorem for the existence of infinitely many braid classes of closed orbits.Comment: Revised version: numerous changes in exposition. Slight modification of two proofs and one definition; 55 pages, 20 figure