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This paper investigates the complex dynamical behavior of a rigid block structure under harmonic ground excitation, thereby mimicking, for instance, the oscillation of the system under seismic excitation or containers placed on a ship under periodic acting of sea waves. The equations of motion are derived, assuming a large frictional coefficient at the interface between the block and the ground, in such a way that sliding cannot occur. In addition, the mathematical model assumes a loss of kinetic energy when an impact with the ground takes place. The resulting mathematical model is then formulated and studied in the framework of impulsive dynamical systems. Its complex dynamical response is studied in detail using two different approaches, based on direct numerical integration and path-following techniques, where the latter is implemented via the continuation platform COCO (Dankowicz and Schilder). Our study reveals the presence of various dynamical phenomena, such as branching points, fold and period-doubling bifurcation of limit cycles, symmetric and asymmetric periodic responses, and chaotic motions. By using the basin stability method, we also investigate the properties of solutions and their ranges of existence in phase and parameter spaces. Moreover, the study considers ground excitation conditions leading to the overturning of the block structure and shows parameter regions wherein such behavior can be avoided.

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We propose a novel technique to analyze multistable, non-linear dynamical systems. It enables one to characterize the evolution of a time-dependent stability margin along stable periodic orbits. By that, we are able to indicate the moments along the trajectory when the stability surplus is minimal, and even relatively small perturbation can lead to a tipping point. We explain the proposed approach using two paradigmatic dynamical systems, i.e., Rössler and Duffing oscillators. Then, the method is validated experimentally using the rig with a double pendulum excited parametrically. Both numerical and experimental results reveal significant fluctuations of sensitivity to perturbations along the considered periodic orbits. The proposed concept can be used in multiple applications including engineering, fluid dynamics, climate research, and photonics.

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A correction to this article has been published and is linked from the HTML version of this paper. The error has been fixed in the paper.

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In this paper we show the first broad experimental confirmation of the basin stability approach. The basin stability is one of the sample-based approach methods for analysis of the complex, multidimensional dynamical systems. We show that investigated method is a reliable tool for the analysis of dynamical systems and we prove that it has a significant advantages which make it appropriate for many applications in which classical analysis methods are difficult to apply. We study theoretically and experimentally the dynamics of a forced double pendulum. We examine the ranges of stability for nine different solutions of the system in a two parameter space, namely the amplitude and the frequency of excitation. We apply the path-following and the extended basin stability methods (Brzeski et al., Meccanica 51(11), 2016) and we verify obtained theoretical results in experimental investigations. Comparison of the presented results show that the sample-based approach offers comparable precision to the classical method of analysis. However, it is much simpler to apply and can be used despite the type of dynamical system and its dimensions. Moreover, the sample-based approach has some unique advantages and can be applied without the precise knowledge of parameter values.

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The subject of the experimental research supported with numerical simulations presented in this paper is an analog electrical circuit representing the ring of unidirectionally coupled single-well Duffing oscillators. The research is concentrated on the existence of the stable three-frequency quasiperiodic attractor in this system. It is shown that such solution can be robustly stable in a wide range of parameters of the system under consideration in spite of a parameter mismatch which is unavoidable during experiment.

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We study the coupling induced destabilization in an array of identical oscillators coupled in a ring structure where the number of oscillators in the ring is large. The coupling structure includes different types of interactions with several next neighbors. We derive an amplitude equation of Ginzburg-Landau type, which describes the destabilization of a uniform stationary state and close-by solutions in the limit of a large number of nodes. Studying numerically an example of unidirectionally coupled Duffing oscillators, we observe a coupling induced transition to collective spatio-temporal chaos, which can be understood using the derived amplitude equations.

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Astrocytes synthesize and release endozepines, a family of regulatory neuropeptides, including diazepam-binding inhibitor (DBI) and its processing fragments such as the octadecaneuropeptide (ODN). At the molecular level, ODN interacts with two types of receptors, i.e. it acts as an inverse agonist of the central-type benzodiazepine receptor (CBR), and as an agonist of a G protein-coupled receptor (GPCR). ODN exerts a wide range of biological effects mediated through these two receptors and, in particular, it regulates astrocyte activity through an autocrine/paracrine mechanism involving the metabotropic receptor. More recently, it has been shown that Müller glial cells secrete phosphorylated DBI and that bisphosphorylated ODN ([bisphospho-Thr(3,9)]ODN, bpODN) has a stronger affinity for CBR than ODN. The aim of the present study was thus to investigate whether bpODN is released by mouse cortical astrocytes and to compare its potency to ODN. Using a radioimmunoassay and mass spectrometry analysis we have shown that bpODN as well as ODN were released in cultured astrocyte supernatants. Both bpODN and ODN increased astrocyte calcium event frequency but in a very different range of concentration. Indeed, ODN stimulatory effect decreased at concentrations over 10(-10)M whereas bpODN increased the calcium event frequency at similar doses. In vivo effects of bpODN and ODN were analyzed in two behavioral paradigms involving either the metabotropic receptor (anorexia) or the CBR (anxiety). As previously described, ODN (100ng, icv) induced a significant reduction of food intake. Similar effect was achieved with bpODN but at a 10 times higher dose (1000 ng, icv). Similarly, and contrasting with our hypothesis, bpODN was also 10 times less potent than ODN to induce anxiety-related behavior in the elevated zero maze test. Thus, the present data do not support that phosphorylation of ODN is involved in receptor selectivity but indicate that it rather weakens ODN activity.

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We study the dynamics of a ring of unidirectionally coupled autonomous Duffing oscillators. Starting from a situation where the individual oscillator without coupling has only trivial equilibrium dynamics, the coupling induces complicated transitions to periodic, quasiperiodic, chaotic, and hyperchaotic behavior. We study these transitions in detail for small and large numbers of oscillators. Particular attention is paid to the role of unstable periodic solutions for the appearance of chaotic rotating waves, spatiotemporal structures, and the Eckhaus effect for a large number of oscillators. Our analytical and numerical results are confirmed by a simple experiment based on the electronic implementation of coupled Duffing oscillators.

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We describe the appearance and stability of spatiotemporal periodic patterns (rotating waves) in unidirectional rings of coupled oscillators with delayed couplings. We show how delays in the coupling lead to the splitting of each rotating wave into several new ones. The appearance of rotating waves is mediated by the Hopf bifurcations of the symmetric equilibrium. We also conclude that the coupling delays can be effectively replaced by increasing the number of oscillators in the chain. The phenomena are shown for the Stuart-Landau oscillators as well as for the coupled FitzHugh-Nagumo systems modeling an ensemble of spiking neurons interacting via excitatory chemical synapses.

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Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be split into two parts: pseudocontinuous and strongly unstable. The pseudocontinuous part of the spectrum mediates destabilization of periodic solutions.

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Synchronization thresholds of an array of nondiagonally coupled oscillators are investigated. We present experimental results which show the existence of ragged synchronizability, i.e., the existence of multiple disconnected synchronization regions in the coupling parameter space. This phenomenon has been observed in an electronic implementation of an array of nondiagonally coupled van der Pol's oscillators. Numerical simulations show good agreement with the experimental observations.

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We have verified some results of Nana and Woafo [Phys. Rev. E 74, 046213 (2006)] in the area of the complete synchronization. We have found that the motion of the van der Pol network is quasiperiodic, not chaotic as the authors have written.