In this paper, the dynamics of the paradigmatic Rössler system is investigated in a yet unexplored region of its three-dimensional parameter space. We prove a necessary condition in this space for which the Rössler system can be chaotic. By using standard numerical tools, like bifurcation diagrams, Poincaré sections, and first-return maps, we highlight both asymptotically stable limit cycles and chaotic attractors. Lyapunov exponents are used to verify the chaotic behavior while random numerical procedures and various plane cross sections of the basins of attraction of the coexisting attractors prove that both limit cycles and chaotic attractors are hidden. We thus obtain previously unknown examples of bistability in the Rössler system, where a point attractor coexists with either a hidden limit cycle attractor or a hidden chaotic attractor.In this paper, a first-order generalized memristor and a polynomial memristor are designed to construct a dual memristive Wien-bridge chaotic system. The proposed system possesses rich dynamic characteristics, including alternating between the periodic state and the chaotic state, variable amplitude and frequency, coexisting attractors, and a locally sustained chaotic state. The dynamic behaviors are obtained and investigated by using Lyapunov exponents, bifurcation diagrams, phase portraits, time-domain waveforms, frequency spectra, and so on. The presented chaotic system is implemented by using a digital signal processing platform. Finally, the National Institute of Standards and Technology test is conducted in this paper. Since the system has rich dynamic behaviors, it has great potential value in encryption engineering fields.We investigate the dynamics of regular fractal-like networks of hierarchically coupled van der Pol oscillators. The hierarchy is imposed in terms of the coupling strengths or link weights. We study the low frequency modes, as well as frequency and phase synchronization, in the network by a process of repeated coarse-graining of oscillator units. At any given stage of this process, we sum over the signals from the oscillator units of a clique to obtain a new oscillating unit. The frequencies and the phases for the coarse-grained oscillators are found to progressively synchronize with the number of coarse-graining steps. Furthermore, the characteristic frequency is found to decrease and finally stabilize to a value that can be tuned via the parameters of the system. We compare our numerical results with those of an approximate analytic solution and find good qualitative agreement. Our study on this idealized model shows how oscillations with a precise frequency can be obtained in systems with heterogeneous couplings. It also demonstrates the effect of imposing a hierarchy in terms of link weights instead of one that is solely topological, where the connectivity between oscillators would be the determining factor, as is usually the case.The detection of an underlying chaotic behavior in experimental recordings is a longstanding issue in the field of nonlinear time series analysis.  https://www.selleckchem.com/products/CP-690550.html  Conventional approaches require the assessment of a suitable dimension and lag pair to embed a given input sequence and, thereupon, the estimation of dynamical invariants to characterize the underlying source. In this work, we propose an alternative approach to the problem of identifying chaos, which is built upon an improved method for optimal embedding. The core of the new approach is the analysis of an input sequence on a lattice of embedding pairs whose results provide, if any, evidence of a finite-dimensional, chaotic source generating the sequence and, if such evidence is present, yield a set of equivalently suitable embedding pairs to embed the sequence. The application of this approach to two experimental case studies, namely, an electronic circuit and magnetoencephalographic recordings of the human brain, highlights how it can make up a powerful tool to detect chaos in complex systems.In the present study, two types of consensus algorithms, including the leaderless coherence and the leader-follower coherence quantified by the Laplacian spectrum, are applied to noisy windmill graphs. Based on the graph construction, exact solutions are obtained for the leader-follower coherence with freely assigned leaders. In order to compare consensus dynamics of two nonisomorphic graphs with the same number of nodes and edges, two generalized windmill graphs are selected as the network models and then explicit expressions of the network coherence are obtained. Then, coherences of models are compared. The obtained results reveal distinct coherence behaviors originating from intrinsic structures of models. Finally, the robustness of the coherence is analyzed. Accordingly, it is found that graph parameters and the number of leaders have a profound impact on the studied consensus algorithms.We investigate the spectral fluctuations and electronic transport properties of chaotic mesoscopic cavities using Kwant, an open source Python programming language based package. Discretized chaotic billiard systems are used to model these mesoscopic cavities. For the spectral fluctuations, we study the ratio of consecutive eigenvalue spacings, and for the transport properties, we focus on Landauer conductance and shot noise power. We generate an ensemble of scattering matrices in Kwant, with desired number of open channels in the leads attached to the cavity. The results obtained from Kwant simulations, performed without or with magnetic field, are compared with the corresponding random matrix theory predictions for orthogonally and unitarily invariant ensembles. These two cases apply to the scenarios of preserved and broken time-reversal symmetry, respectively. In addition, we explore the orthogonal to unitary crossover statistics by varying the magnetic field and examine its relationship with the random matrix transition parameter.Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts-empirical mode decomposition (EMD) and generalized fractal dimensions-into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Hénon map, the Lorenz '63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz '96 model and the on-off intermittency model).