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Paul M Riechers, James P Crutchfield
The meromorphic functional calculus developed in Part I overcomes the nondiagonalizability of linear operators that arises often in the temporal evolution of complex systems and is generic to the metadynamics of predicting their behavior. Using the resulting spectral decomposition, we derive closed-form expressions for correlation functions, finite-length Shannon entropy-rate approximates, asymptotic entropy rate, excess entropy, transient information, transient and asymptotic state uncertainties, and synchronization information of stochastic processes generated by finite-state hidden Markov models...
March 2018: Chaos
K Premalatha, V K Chandrasekar, M Senthilvelan, M Lakshmanan
We investigate the occurrence of collective dynamical states such as transient amplitude chimera, stable amplitude chimera, and imperfect breathing chimera states in a locally coupled network of Stuart-Landau oscillators. In an imperfect breathing chimera state, the synchronized group of oscillators exhibits oscillations with large amplitudes, while the desynchronized group of oscillators oscillates with small amplitudes, and this behavior of coexistence of synchronized and desynchronized oscillations fluctuates with time...
March 2018: Chaos
Oleg V Maslennikov, Vladimir I Nekorkin, Jürgen Kurths
We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions...
March 2018: Chaos
Nastaran Lotfi, Francisco A Rodrigues, Amir Hossein Darooneh
In this paper, we analyze explosive synchronization in networks with a community structure. The results of our study indicate that the mesoscopic structure of the networks could affect the synchronization of coupled oscillators. With the variation of three parameters, the degree probability distribution exponent, the community size probability distribution exponent, and the mixing parameter, we could have a fast or slow phase transition. Besides, in some cases, we could have communities which are synchronized inside but not with other communities and vice versa...
March 2018: Chaos
Gabriela I Depetri, Felipe A C Pereira, Boris Marin, Murilo S Baptista, J C Sartorelli
The dynamics of a parametric simple pendulum submitted to an arbitrary angle of excitation ϕ was investigated experimentally by simulations and analytically. Analytical calculations for the loci of saddle-node bifurcations corresponding to the creation of resonant orbits were performed by applying Melnikov's method. However, this powerful perturbative method cannot be used to predict the existence of odd resonances for a vertical excitation within first order corrections. Yet, we showed that period-3 resonances indeed exist in such a configuration...
March 2018: Chaos
Paul M Riechers, James P Crutchfield
Virtually all questions that one can ask about the behavioral and structural complexity of a stochastic process reduce to a linear algebraic framing of a time evolution governed by an appropriate hidden-Markov process generator. Each type of question-correlation, predictability, predictive cost, observer synchronization, and the like-induces a distinct generator class. Answers are then functions of the class-appropriate transition dynamic. Unfortunately, these dynamics are generically nonnormal, nondiagonalizable, singular, and so on...
March 2018: Chaos
Marcus Brandenburg
Managing supply networks is a highly relevant task that strongly influences the competitiveness of firms from various industries. Designing supply networks is a strategic process that considerably affects the structure of the whole network. In contrast, supply networks for new products are configured without major adaptations of the existing structure, but the network has to be configured before the new product is actually launched in the marketplace. Due to dynamics and uncertainties, the resulting planning problem is highly complex...
March 2018: Chaos
Susmita Sadhu, Christian Kuehn
The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise...
March 2018: Chaos
Varun Pandit, Archan Mukhopadhyay, Sagar Chakraborty
Replicator equation-a paradigm equation in evolutionary game dynamics-mathematizes the frequency dependent selection of competing strategies vying to enhance their fitness (quantified by the average payoffs) with respect to the average fitnesses of the evolving population under consideration. In this paper, we deal with two discrete versions of the replicator equation employed to study evolution in a population where any two players' interaction is modelled by a two-strategy symmetric normal-form game. There are twelve distinct classes of such games, each typified by a particular ordinal relationship among the elements of the corresponding payoff matrix...
March 2018: Chaos
Dawid Dudkowski, Awadhesh Prasad, Tomasz Kapitaniak
We study the concepts of regular and perpetual points for describing the behavior of chaotic attractors in dynamical systems. The idea of these points, which have been recently introduced to theoretical investigations, is thoroughly discussed and extended into new types of models. We analyze the correlation between regular and perpetual points, as well as their relation with phase space, showing the potential usefulness of both types of points in the qualitative description of co-existing states. The ability of perpetual points in finding attractors is indicated, along with its potential cause...
March 2018: Chaos
Sergey A Vakulenko, Ivan Sudakov, Luke Mander
In this paper, we study a model of many species that compete, directly or indirectly, for a pool of common resources under the influence of periodic, stochastic, and/or chaotic environmental forcing. Using numerical simulations, we find the number and sequence of species going extinct when the community is initially packed with a large number of species of random initial densities. Thereby, any species with a density below a given threshold is regarded to be extinct.
March 2018: Chaos
D Ibáñez-Soria, J Garcia-Ojalvo, A Soria-Frisch, G Ruffini
Generalized synchronization between coupled dynamical systems is a phenomenon of relevance in applications that range from secure communications to physiological modelling. Here, we test the capabilities of reservoir computing and, in particular, echo state networks for the detection of generalized synchronization. A nonlinear dynamical system consisting of two coupled Rössler chaotic attractors is used to generate temporal series consisting of time-locked generalized synchronized sequences interleaved with unsynchronized ones...
March 2018: Chaos
Frank Kwasniok
A data-driven linear framework for detecting, anticipating, and predicting incipient bifurcations in spatially extended systems based on principal oscillation pattern (POP) analysis is discussed. The dynamics are assumed to be governed by a system of linear stochastic differential equations which is estimated from the data. The principal modes of the system together with corresponding decay or growth rates and oscillation frequencies are extracted as the eigenvectors and eigenvalues of the system matrix. The method can be applied to stationary datasets to identify the least stable modes and assess the proximity to instability; it can also be applied to nonstationary datasets using a sliding window approach to track the changing eigenvalues and eigenvectors of the system...
March 2018: Chaos
Yuval R Zelnik, Punit Gandhi, Edgar Knobloch, Ehud Meron
Many ecosystems show both self-organized spatial patterns and multistability of possible states. The combination of these two phenomena in different forms has a significant impact on the behavior of ecosystems in changing environments. One notable case is connected to tristability of two distinct uniform states together with patterned states, which has recently been found in model studies of dryland ecosystems. Using a simple model, we determine the extent of tristability in parameter space, explore its effects on the system dynamics, and consider its implications for state transitions or regime shifts...
March 2018: Chaos
Yoshito Hirata
I propose a method for reconstructing multi-dimensional dynamical noise inspired by the embedding theorem of Muldoon et al. [Dyn. Stab. Syst. 13, 175 (1998)] by regarding multiple predictions as different observables. Then, applying the embedding theorem by Stark et al. [J. Nonlinear Sci. 13, 519 (2003)] for a forced system, I produce time series forecast by supplying the reconstructed past dynamical noise as auxiliary information. I demonstrate the proposed method on toy models driven by auto-regressive models or independent Gaussian noise...
March 2018: Chaos
Nevin Thomas, Sirshendu Mondal, Samadhan A Pawar, R I Sujith
We here systematically investigate amplitude death (AD) phenomenon in a thermoacoustic system using a mathematical model of coupled prototypical thermoacoustic oscillators, the horizontal Rijke tubes. AD has recently been identified as a relatively simple phenomenon, which can be exploited to stop the unwanted high amplitude pressure oscillations resulting from the occurrence of thermoacoustic instability. We examine the effect of time-delay and dissipative couplings on a system of two Rijke tubes when they are symmetrically and asymmetrically coupled...
March 2018: Chaos
Nicolás Rubido, Celso Grebogi, Murilo S Baptista
Finding the correct encoding for a generic dynamical system's trajectory is a complicated task: the symbolic sequence needs to preserve the invariant properties from the system's trajectory. In theory, the solution to this problem is found when a Generating Markov Partition (GMP) is obtained, which is only defined once the unstable and stable manifolds are known with infinite precision and for all times. However, these manifolds usually form highly convoluted Euclidean sets, are a priori unknown, and, as it happens in any real-world experiment, measurements are made with finite resolution and over a finite time-span...
March 2018: Chaos
Maria Obregon, Nawin Raj, Yury Stepanyants
The adiabatic decay of different types of internal wave solitons caused by the Earth's rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions: (ut  + c ux  + α u ux  + α1  u2  ux  + β uxxx )x  = γ u. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth's rotation described by the parameter γ...
March 2018: Chaos
A Anzo-Hernández, H E Gilardi-Velázquez, E Campos-Cantón
We present dissipative systems with unstable dynamics called the unstable dissipative systems which are capable of generating a multi-stable behavior, i.e., depending on its initial condition, the trajectory of the system converges to a specific attractor. Piecewise linear (PWL) systems are generated based on unstable dissipative systems, whose main attribute when they are switched is the generation of chaotic trajectories with multiple wings or scrolls. For this PWL system, a structure is proposed where both the linear part and the switching function depend on two parameters...
March 2018: Chaos
Zhongkui Sun, Rui Xiao, Xiaoli Yang, Wei Xu
Oscillation quenching has been widely studied during the past several decades in fields ranging from natural sciences to engineering, but investigations have so far been restricted to oscillators with an integer-order derivative. Here, we report the first study of amplitude death (AD) in fractional coupled Stuart-Landau oscillators with partial and/or complete conjugate couplings to explore oscillation quenching patterns and dynamics. It has been found that the fractional-order derivative impacts the AD state crucially...
March 2018: Chaos
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