Anharmonic interactions in lattices may sustain robust oscillatory modes and (nonlinear) waves including solitons. This is illustrated here by using an exponentially repulsive interaction introduced by Toda. To cope with friction and dissipation -always present in real systemsand hence to make robust, e.g., solitons, following Lord Rayleigh, an appropriate input-output energy balance is added t...

In this paper, hyperclone lattice is studied via three kinds of embeddings. One is from the clone lattice on A to the hyperclone lattice on A, the second is from the hyperclone lattice on A to the clone lattice on P (A) \ {∅} and the third one is from the hyperclone lattice on A′ to the hyperclone lattice on A, for A′ ⊂ A and the finite set A. The second map has usually been used for the descri...

In this paper, we consider a lattice of the coupled logistic map with periodic boundary conditions. We prove that synchronization occurs in the one-dimensional lattice with lattice size n = 4 for any γ in the chaotic regime [γ∞ ≈ 3.57, 4]. It is worthwhile to emphasize that, despite of the fact that there is a rigorous proof for synchronization in many systems with continuous time, almost nothi...

We introduce the Hierarchically Growing Hyperbolic Self-Organizing Map (H2SOM) featuring two extensions of the HSOM (hyperbolic SOM): (i) a hierarchically growing variant that allows for incremental training with an automated adaptation of lattice size to achieve a prescribed quantization error and (ii) an approximate best match search that utilizes the special structure of the hyperbolic latti...

For P a poset or lattice, let Id(P ) denote the poset, respectively, lattice, of upward directed downsets in P, including the empty set, and let id(P ) = Id(P )−{∅}. This note obtains various results to the effect that Id(P ) is always, and id(P ) often, “essentially larger” than P. In the first vein, we find that a poset P admits no <-respecting map (and so in particular, no one-to-one isotone...

A novel approach to the determination of the neighbourhood and the identification of spatio-temporal dynamical systems is investigated. It is shown that thresholding to convert the pattern to a binary pattern and then applying cellular automata (CA) neighbourhood detection methods can provide an initial estimate of the neighbourhood. A coupled map lattice model can then be identified using the ...

We study the phase diagram of the sine circle map lattice with random initial conditions and identify the various types of dynamical behaviour which appear here. We focus on the regions which show spatio-temporal intermittency and characterise the accompanying scaling behaviour. Directed percolation exponents are seen at some points in the parameter space in the neighbourhood of bifurcation bou...

We introduce the Hierarchically Growing Hyperbolic Self-Organizing Map (H2SOM) featuring two extensions of the HSOM (hyperbolic SOM): (i) a hierarchically growing variant that allows for incremental training with an automated adaptation of lattice size to achieve a prescribed quantization error and (ii) an approximate best match search that utilizes the special structure of the hyperbolic latti...