Use of False Nearest Neighbours for Selecting Variables and Embedding Parameters for State Space Reconstruction

If data are generated by a system with a d-dimensional attractor, then Takens’ theorem guarantees that reconstruction that is diffeomorphic to the original attractor can be built from the single time series in 2d+1-dimensional phase space. However, under certain conditions, reconstruction is possible even in a space of smaller dimension. This topic is very important because the size of the reconstruction space relates to the effectiveness of the whole subsequent analysis. In this paper, the false nearest neighbour (FNN) methods are revisited to estimate the optimum embedding parameters and the most appropriate observables for state space reconstruction. A modification of the false nearest neighbour method is introduced. The findings contribute to evidence that the length of the embedding time window (TW) is more important than the reconstruction delay time and the embedding dimension (ED) separately. Moreover, if several time series of the same system are observed, the choice of the one that is used for the reconstruction could also be critical. The results are demonstrated on two chaotic benchmark systems.

Some Fixed Point Theorems in Complex Valued b-Metric Spaces

Recently, Azam et al. introduced the notion of complex valued metric spaces and proved fixed point theorems under the contraction condition. Rao et al. introduced the notion of complex valued b-metric spaces. In this paper, we obtain some fixed point results for the mapping satisfying rational expressions in complex valued b-metric spaces. Also, an example is given to illustrate our obtained result.

Studying the Relationship between System-Level and Component-Level Resilience

The capacity to maintain stability in a system relies on the components which make up the system. This study explores the relationship between component-level resilience and system-level resilience with the aim of identifying policies which foster system-level resilience in situations where existing incentives might undermine it. We use an abstract model of interacting specialized resource users and producers which can be parameterized to represent specific real systems. We want to understand how features, such as stockpiles, influence system versus component resilience. Systems are subject to perturbations of varying intensity and frequency. For our study, we create a simplified economy in which an inventory carrying cost is imposed to incentivize smaller inventories and examine how components with varying inventory levels compete in environments subject to periods of resource scarcity. The results show that policies requiring larger inventories foster higher component-level resilience but do not foster higher system-level resilience. Inventory carrying costs reduce production efficiency as inventory sizes increase. JIT inventory strategies improve production efficiency but do not afford any buffer against future uncertainty of resource availability.

Group Measures and Modeling for Social Networks

Social network modeling is generally based on graph theory, which allows for study of dynamics and emerging phenomena. However, in terms of neighborhood, the graphs are not necessarily adapted to represent complex interactions, and the neighborhood of a group of vertices can be inferred from the neighborhoods of each vertex composing that group. In our study, we consider that a group has to be considered as a complex system where emerging phenomena can appear. In this paper, a formalism is proposed to resolve this problematic by modeling groups in social networks using pretopology as a generalization of the graph theory. After giving some definitions and examples of modeling, we show how some measures used in social network analysis (degree, betweenness, and closeness) can be also generalized to consider a group as a whole entity.

Dynamic Cournot Duopoly Game with Delay

The delay Cournot duopoly game is studied. Dynamical behaviors of the game are studied. Equilibrium points and their stability are studied. The results show that the delayed system has the same Nash equilibrium point and the delay can increase the local stability region.

Conception of Biologic System: Basis Functional Elements and Metric Properties

A notion of biologic system or just a system implies a functional wholeness of comprising system components. Positive and negative feedback are the examples of how the idea to unite anatomical elements in the whole functional structure was successfully used in practice to explain regulatory mechanisms in biology and medicine. There are numerous examples of functional and metabolic pathways which are not regulated by feedback loops and have a structure of reciprocal relationships. Expressed in the matrix form positive feedback, negative feedback, and reciprocal links represent three basis elements of a Lie algebra sl(2,ℝ) of a special linear group SL(2,ℝ). It is proposed that the mathematical group structure can be realized through the three regulatory elements playing a role of a functional basis of biologic systems. The structure of the basis elements endows the space of biological variables with indefinite metric. Metric structure resembles Minkowski's space-time (+, −, −) making the carrier spaces of biologic variables and the space of transformations inhomogeneous. It endows biologic systems with a rich functional structure, giving the regulatory elements special differentiating features to form steady autonomous subsystems reducible to one-dimensional components.

Complex Stochastic Boolean Systems: Comparing Bitstrings with the Same Hamming Weight

A complex stochastic Boolean system (CSBS) is a complex system depending on an arbitrarily large number n of random Boolean variables. CSBSs arise in many different areas of science and engineering. A proper mathematical model for the analysis of such systems is based on the intrinsic order: a partial order relation defined on the set 0,1n of all binary n-tuples of 0s and 1s. The intrinsic order enables one to compare the occurrence probabilities of two given binary n-tuples with no need to compute them, simply looking at the relative positions of their 0s and 1s. Regarding the analysis of CSBSs, the intrinsic order reduces the complexity of the problem from exponential (2n binary n-tuples) to linear (n Boolean variables). In this paper, using the intrinsic ordering, we compare the occurrence probabilities of any two binary n-tuples having the same number of 1-bits (i.e., the same Hamming weight). Our results can be applied to any CSBS with mutually independent Boolean variables.

A Small Morris-Lecar Neuron Network Gets Close to Critical Only in the Small-World Regimen

Spontaneous emergence of neuronal activity avalanches characterized by power-law distributions is known to occur in different types of nervous tissues suggesting that nervous systems may operate at a critical regime. Here, we explore the possible relation of this dynamical state with the underlying topology in a small-size network of interconnected Morris-Lecar neurons. Studying numerically different topological configurations, we find that, very close to the efficient small-world situation, the system self-organizes near to a critical branching process with observable distributions in the proximity of a power law with exponents similar to those reported in the experimental literature. Therefore, we conclude that the observed scaling is intimately related only with the small-world topology.

Weakly Compatible Maps Using E.A. and (CLR) Properties in Complex Valued G-Metric Spaces

We introduce the notion of complex valued G-metric spaces and prove common fixed point theorems for weakly compatible maps along with E.A. and (CLR) properties in complex valued G-metric spaces.

Common Fixed Point Theorems for a Rational Inequality in Complex Valued Metric Spaces

We prove a common fixed point theorem for a pair of mappings. Also, we prove a common fixed point theorem for pairs of self-mappings along with weakly commuting property.