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Experimental observations and numerical simulations of synchronization and antisynchronization of low-frequency power drop-outs and jump-ups of chaotic semiconductor lasers are presented.

We address the question of bounds on the synchronization error for the case of nearly identical nonlinear systems. It is pointed out that negative largest conditional Lyapunov exponents of the synchronization manifold are not sufficient to guarantee a small synchronization error and that one has to find bounds for the deformation of the manifold due to perturbations. We present an analytic bound for a simple subclass of systems, which includes the Lur'e systems, showing that the bound for the deformation grows as the largest singular value of the linearized system gets larger. Then, the Lorenz system is taken as an example to demonstrate that the phenomenon is not restricted to Lur'e systems.

Experimental observations of synchronization and anti‐synchronization of low‐frequency‐fluctuations of external cavity semiconductor lasers are presented. Antisynchronization occures not only for the well known power drop‐out dynamic but also for the power jump‐up dynamic. Detailed experimental investigations of the parameter space reveal several seperate areas of (anti‐)synchronization. Numerical simulations with the Lang‐Kobayashi model equations are presented for the antisynchronized power drop‐out case.

A systematic coupling procedure is introduced for synchronizing arbitrary chaotic dynamical systems. This coupling exploits the existing contraction properties of the flow and suppresses divergence only along those directions in state space, where the underlying flow is not contracting. In this way, systems can be synchronized using a minimum of transmitted information for guaranteed high-quality synchronization. Applications in combination with sporadic driving and in partitioned state spaces are numerically illustrated.

Using the example of periodically driven multi-bubble systems we demonstrate that it may be misleading to conclude from a successful low dimensional delay reconstruction that the underlying dynamical process is indeed governed by a low-dimensional attractor.

The occurrence of phase synchronization of a pair of unidirectionally coupled nonidentical Ginzburg-Landau equations is demonstrated and characterized using cyclic and extended phases. Furthermore, it is shown that weak coupling first leads to frequency synchronization and later to phase synchronization. For strong coupling there is evidence for generalized synchronization.

Different synchronization phenomena of coupled chaotic systems are presented and illustrated by numerical simulations. The examples given include not only low dimensional chaos but also hyperchaotic spatially extended systems. Furthermore, the notions of generalized synchronization and phase synchronization are introduced that may be used to study coupled nonidentical systems.

The review gives and account of the historical development, the current state and possible future developments of experimental nonlinear physics, with emphasis on acoustics, hydrodynamics and optics. The concepts of nonlinear time-series analysis which are the basis of the analysis of experimental outcomes from nonlinear systems are explained and recent developments pertaining to such different fields as modeling, prediction, nonlinear noise reduction, detecting determinism, synchronization, and spatio-temporal time series are surveyed. An overview is given of experiments on acoustic cavitation, a field rich of nonlinear phenomena such as nonlinear oscillations, chaotic dynamics and structure formation, and one of the first physical systems to exhibit period-doubling and chaos in experiment.

Spatiotemporal time series are analyzed and predicted using reconstructed local states. As numerical examples the evolution of a Kuramoto-Sivashinsky equation and a coupled map lattice are predicted from previously sampled data.

We present methods for evaluating bifurcation diagrams that were reconstructed from time series generated by the system of interest at different fixed parameter values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

During optical breakdown, the energy delivered to the sample is either transmitted, reflected, scattered, or absorbed. The absorbed energy can be further divided into the energy required to evaporate the focal volume, the energy radiated by the luminescent plasma, and the energy contributing to the mechanical effects such as shock wave emission and cavitation. The partition of the pulse energy between these channels was investigated for 4 selected laser parameters (6 ns pulses of 1 and 10 mJ, 30 ps pulses of 50 (mu) J and 1 mJ, all at 1064 nm). The results indicated that the scattering and reflection by the plasma is small compared to plasma transmission. The plasma absorption can therefore be approximated by A approximately equals (1-T). The ratio of the shock wave energy and cavitation bubble energy was found to be approximately constant (between 1.5:1 and 2:1). For a more comprehensive study of the influence of pulse duration and focusing angle on the energy partition, we therefore restricted our measurements to the plasma transmission and the cavitation bubble energy. The bubble energy was used as an indicator for the total amount of mechanical energy produced. We found that the plasma absorption first decreases strongly with decreasing pulse duration, but increases again for pulses shorter than 3 ps. The conversion of the absorbed energy into mechanical energy is approximately equal to 90% with ns-pulses at large focusing angles. It decreases both with decreasing focusing angle and pulse duration (to less than or equal to 15% for fs-pulses). The disruptive character of plasma-mediated laser surgery is therefore reduced with ultrashort laser pulses.

This tutorial review presents an overview of the achievements and some present research activities in the field of state space based methods for nonlinear time-series analysis. In particular, questions of state space reconstruction, of modelling and prediction, of filtering and noise reduction, of detecting non-linearities in time series, and applications using chaotic synchronization are addressed. Furthermore, a new approach for modeling data from spatio-temporal systems is presented.

329 Superresolution analysis identifies proliferative changes of excitable membranes (T-tubules) during cardiac remodeling leading to orphaning of Ca2+ release sites and prolonged action potentials {#article-title-2} Purpose: Membrane invaginations called transverse tubules (TTs) are essential to couple cellular depolarization to thousands of intracellular Ca2+ release units (CRUs) in ventricular myocytes. While confocal studies suggested loss of TTs as a leading mechanism in heart failure (HF), membrane invaginations have nanometric dimensions which are not resolved by conventional light microscopy. We employed STED superresolution nanoscopy and quantitative analysis strategies which directly address TT changes in different HF models to characterize remodeling mechanisms within clinically relevant time scales over weeks. Methods: Novel superresolution microscopy strategies for TTs in living myocytes and multimodal confocal imaging of TTs and Ca2+ release were used. Thereby, we characterized structural and functional changes as well as different stages of TT remodeling. Both post-MI and post-TAC murine models of HF were investigated early (1 week), intermediate (4 weeks) and late (8 weeks) after intervention versus sham. Computational modeling of 20,000 intracellular CRUs was used to analyze heterogeneous cellular changes to predict the macroscopic outcome for the cell-wide Ca2+ transient and the action potential (AP). Results: STED showed that the mean area of individual TT cross-sections increased significantly starting early, and progressing through 4 weeks to 8 weeks during HF development. Interestingly, the radius size distribution of individual TTs was significantly right-shifted including a second (abnormal) peak which represent distinct, enlarged membrane structures consistent with heterogeneous membrane changes. The cell-wide TT network showed significant total increases of longitudinal components and branching complexity. The CRU associated protein junctophilin-2 was significantly downregulated, whereas ryanodine receptor (RyR2) clusters were increasingly displaced from striations, suggesting heterogeneous orphaning mechanisms of Ca2+ release sites in HF. Computational modeling of increased heterogeneity showed delayed Ca2+ release early and late in diastole, leading to AP prolongation and significant diastolic Ca2+ leak. Conclusions: We identify proliferative mechanisms of TT remodeling in two independent HF models (post-MI; post-TAC). Our data support local subcellular mechanisms of TT reorganization resulting in increasingly heterogeneous membrane structures and defective CRUs including orphaning. The combined TT and CRU changes may directly contribute to dyssynchronous Ca2+ release and AP prolongation, and further set the propensity for afterdepolarizations.

The parameters of a given (chaotic) dynamical model are estimated from scalar time series by adapting a computer model until it synchronizes with the given data. This parameter identification method is applied to numerically generated and experimental data from Chua's circuit.

The synchronization of spatiotemporal chaos of two arrays of coupled nonlinear oscillators is achieved by discrete time coupling of individual cells of the arrays. This synchronization method is based on the knowledge of the local dynamics and can be applied to any type of arrays where the synchronization properties of the cells are known. Furthermore, we discuss possible applications of synchronizing spatiotemporal chaos in communication and anticontrol of chaos.

We discuss a general approach for chaotic synchronization of dynamical systems that is based on an active-passive decomposition (APD) of given dynamical systems. It is shown how this approach can be used to construct high-dimensional synchronizing systems in a systematic way using low-dimensional systems as building blocks. Furthermore, two methods for encoding messages are considered that are both based on synchronization. Using these methods the quality of the reconstructed information signal is higher and the encoding is more secure compared to other encryption methods based on synchronization. The main ideas are illustrated using experimental and numerical examples based on continuous and discrete dynamical systems.

Publisher Summary A general approach for constructing chaotic synchronized dynamical systems is discussed that is based on a decomposition of given systems into active and passive parts. As a possible application the chapter considers an improved encoding method where the information signal is injected into the dynamical system of the transmitter. Furthermore, it highlights how to design in a systematic way high-dimensional synchronized systems that may be used for efficient hyperchaotic encoding of information. Synchronization of periodic signals is a well-known phenomenon in physics, engineering, and many other scientific disciplines.

In super-resolution microscopy applications, previously unknown fluorescent signal patterns are recorded. The interpretation of such nanoscale data is often unexpectedly complex, and can be performed by different analysis strategies: 1) empirical statistics of the spatial distribution of intensity values to identify local objects, 2) inverse problem approaches to convert signals into objects based on external data models or other external assumptions, 3) decomposition of spatial signal patterns into spatial modes.Here, we compare and relate two distinct approaches of analysing STED microscopy images of RyR2 clusters (cardiac ryanodine receptor type 2). RyR2 Ca2+ release channels are essential for heart muscle function (excitation-contraction coupling). Yet, the nature of lateral channel organization within super-structural clusters is unknown and important for models of local control mechanisms of RyR2 Ca2+ release activity.We established multi-scale analysis of RyR2 signal patterns employing wavelet analysis. This analysis decomposes the initial image with predefined wavelets into spatial modes identifying dominant scales of signal fluctuations. We tested the sensitivity of this approach for different wavelets with artificial and modified images. Distinct scales represent inter-cluster spacing and intra-cluster patterns, respectively. Furthermore, we compare the spatial mode analysis with object-based approaches. For object-based analysis, RyR2 cluster sub-structures were identified with a multi-step thresholding procedure. After increasing the threshold level step-by-step, the hierarchy of the segmentation output was analysed with logical operators. Accordingly, we identified cluster sub-structures as discrete objects of variable sizes with typical spacings ranging from 78 to 128 nm (IQR range) that we interpret as individual cluster building-blocks.We conclude that identification of common protein cluster building principles in highly variable signal structures as typical for RyR2 clusters benefits from combining object-based approaches with spatial mode analysis.

Conditions for the occurrence of generalized synchronization of unidirectionally coupled dynamical systems are given in terms of asymptotic stability. All theoretical results are illustrated by analytical and numerical examples.© (1995) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

We present experimental investigations and numerical calculation of the shock wave emission and cavitation bubble expansion after optical breakdown in water with Hd:YAG laser pulses of 30 ps and 6 ns duration. The experimental investigations were done by time-resolved photography with a time resolution of 30 ps, or 6 ns, respectively, and a spatial resolution of 4 micrometers . Position and velocity of both the shock front and the bubble wall were determined, and the shock wave pressure p(r) was then calculated from the shock velocity. Calculations of the bubble formation and shock wave emission were performed using the Gilmore model of cavitation bubble dynamics. The calculations yield the dynamics of the bubble wall, the pressure evolution p(t) inside the bubble, and pressure profiles in the surrounding liquid at fixed times after the start of the laser pulse. The maximal shock wave pressure was measured to be 2400 MPa after a 1 mJ ns-pulse, and 1700 MPa after a ps-pulse of the same energy. The initial shock wave duration was slightly shorter for the ns-pulse than for the ps-pulse, and had a (calculated) value of 30 ns and 46 ns, respectively. Due to nonlinear effects, the duration increased to about 75 ns (measured for the ps-pulse) during propagation of the first few millimeters. A formation phase of the shock front was observed after the ns-pulse, but not after the ps-pulse, where the shock front arose within less than 100 ps after the end of the laser pulse. After shock front formation, the pressure decay was approximately proportional to r -2 . The maximal bubble wall velocity was 1850 m/s after the 1 mJ ns-pulse, and 780 m/s after the ps-pulse. In general, good agreement was observed between the results of the calculations and the experimental data. The Gilmore model is therefore well suited to calculate the shock wave emission and the initial phase of bubble expansion after laser-induced plasma generation. Since it can cover a wide parameter range, it may serve as a tool for the optimization of laser parameters in medical laser applications.

The transmission of digital signals by means of chaotic synchronization is demonstrated, numerically as well as experimentally, via Chua’s circuit.

Secure communications via chaotic synchronization is experimentally demonstrated using Chua's circuit. In our experiment the energy lost in the information-bearing signal is approximately 0.6 dBV. The reduction in chaotic signal after the recovery process is between -40 and -50 dBV.

Synchronization scenarios of coupled mechanical metronomes are studied by means of numerical simulations showing the onset of synchronization for two, three, and 100 globally coupled metronomes in terms of Arnol'd tongues in parameter space and a Kuramoto transition as a function of coupling strength. Furthermore, we study the dynamics of metronomes where overturning is possible. In this case hyperchaotic dynamics associated with some diffusion process in configuration space is observed, indicating the potential complexity of metronome dynamics.

Several methods are examined which allow to produce forecasts for time series in the form of probability assignments. The necessary concepts are presented, addressing questions such as how to assess the performance of a probabilistic forecast. A particular class of models, cluster weighted models (CWMs), is given particular attention. CWMs, originally proposed for deterministic forecasts, can be employed for probabilistic forecasting with little modification. Two examples are presented. The first involves estimating the state of (numerically simulated) dynamical systems from noise corrupted measurements, a problem also known as filtering. There is an optimal solution to this problem, called the optimal filter, to which the considered time series models are compared. (The optimal filter requires the dynamical equations to be known.) In the second example, we aim at forecasting the chaotic oscillations of an experimental bronze spring system. Both examples demonstrate that the considered time series models, and especially the CWMs, provide useful probabilistic information about the underlying dynamical relations. In particular, they provide more than just an approximation to the conditional mean.

The bifurcation sets of driven strictly dissipative nonlinear oscillators are compared in terms of phase diagrams and fixed-point diagrams. The comparison reveals distinctive bifurcation patterns that occur for all models. In particular, two subpatterns of bifurcation curves within the period-doubling hierarchy can be identified. The results suggest that there exists a universal bifurcation structure for oscillators of the type investigated.

For a sparsely connected and balanced network with unknown interaction delays and 1D elements, we show that driving the network to steady states can reveal its cooperative architecture, i.e., the dynamics and output functions of individual nodes and the connection topology. Our approach is robust and can be extended to (sub)networks with a small topology uncertainty. Some examples are presented to illustrate the proposed identification method.

Stabilization of chaotic intensity fluctuations of intracavity frequency-doubled solid-state (Nd: YAG) lasers using multiple-delay feedback control (MDFC) is demonstrated by numerical simulations. It is shown that MDFC not only provides stable (cw) output for constant pump rates but also works with slowly varying pump currents, resulting in corresponding (nonchaotic) intensity modulations.

Acoustical cavitation bubble fields exhibit spatio–temporal structure formation on multiple scales involved. The temporal scales are the fast, medium, and slow scales given by the period of the driving, the typical relaxation time of the individual bubble motion, and the drift of the filaments, respectively. The spatial scales are the microscopic, mesoscopic, and macroscopic scales given by the bubbles’ mean radii, typical bubble distances, and the wavelength of macroscopic disturbances (e.g., sound waves). Prominent examples of the observed structures are, for example, period doubling on a fast temporal scale and a slowly drifting filamentlike bubble distribution on the macroscopic spatial scale. These structures are experimentally investigated using simultaneously measured sequences of images (CCD camera) and acoustic time series (hydrophones). Based on these data the experimentally observed patterns are characterized using image processing techniques (e.g., principal component analysis) and by means of...

For bubble oscillations under medium and large pressure amplitudes a complicated scenario of bifurcations and coexisting (chaotic) attractors exists. However, for very small bubbles in a very strong sound field the dynamics becomes regular and a new type of strong resonance with a thresholdlike increase in oscillation amplitude occurs [E. A. Neppiras and B. E. Noltingk, Proc. Phys. Soc. London Ser. B 64, 1032 (1951); H. G. Flynn, in Physical Acoustics Vol. 1, edited by W. P. Mason (1964), p. 57; W. Lauterborn, Acustica 20, 105 (1968)]. This phenomenon has a strong influence on many properties of cavitation bubbles. The following aspects are considered: rectified diffusion, stability of the bubble under SBSL conditions, primary and secondary Bjerknes forces, and interpretation of bubble size measurements.

This article gives an introduction to the research area of chaos physics. The new language and the basic tools are presented and illustrated by examples from acoustics: a bubble in water driven by a sound field and other nonlinear oscillators. The notions of strange attractors and their basins, bifurcations and bifurcation diagrams, Poincare maps, phase diagrams, fractal dimensions, scaling spectra, reconstruction of attractors from time series, winding numbers, as well as Lyapunov exponents, spectra, and diagrams are addressed.

Resonance curves, bifurcation diagrams, and phase diagrams of the Duffing equation x + d x + x + x 3 = f cos (ωt) are presented. They show a periodic recurrence of a specific fine structure in the bifurcation set, which is closely connected with the nonlinear resonances of the system.

Bifurcation diagrams of the driven van der Pol oscillator are given showing mode-locking and period-doubling cascades. At low driving amplitudes locking regions occur following Farey sequences. At high driving amplitudes this relationship is destroyed due to the appearance of period-doubling cascades and coexisting attractors. A generalization of the winding number is used to compute devil's staircases and winding-number diagrams of period-doubling cascades. The winding numbers at the period-doubling bifurcation points constitute an alternating sequence that converges at the accumulation point of the cascade.

Two unidirectionally coupled external cavity semiconductor lasers showing chaotic intensity fluctuations are studied by numerically solving the Lang-Kobayashi model equations [IEEE J. Quantum Electron. QE-16, 347 (1980)]. The systems are shown to synchronize when operating in the regime of low-frequency fluctuations, which is characterized by a very high-dimensional $(dg150)$ attractor. The influence of parameter differences between the two lasers on the synchronization quality is investigated.

Mechanical and functional behaviors of cardiac myocytes are largely determined by architectural arrangements of protein assemblies, membrane networks and organelles. Myofilaments and mitochondria fill most of the cell volume and are wrapped by the endomembrane systems that include the endo(sarco)plasmic reticulum (ESR) that is contiguous with the nuclear membrane system. While these membrane systems and organelles are essential for metabolic, proteomic and energetic homeostasis, the ESR further forms extensive contacts with the branches of the internal sarcolemmal membrane network, the T-tubules (TTs), and interacts with the cytoskeleton. In particular, TTs distribute electrical and chemical signals to intracellular Ca2+ release nanodomains that involve junctional ESR domains. We analyze the TT network quantitatively deep inside living myocytes based on data from confocal and super-resolution (STED) imaging. In adult murine cells, rectilinear TT elements showed a bimodal distribution of longitudinal and transversal orientations suggesting regular network properties. Furthermore, the properties of network branches corresponded with regular network architectures. In contrast, 4 weeks after myocardial infarction (post-MI) the orientations changed significantly, showing a differential increase versus decrease of longitudinal and transversal elements, respectively. Importantly, the number of branch points and oblique elements increased significantly post-MI. Thus, large-scale rectilinear network organization may support unique physiological functions, which become reorganized post-MI, leading to increased network complexity and dysfunction. It remains unclear how TTs interface with the microtubule (MT) system which is associated with junctional and other ESR structures, cortical scaffolds, and protein trafficking. Therefore, we further examined the MT network architecture which will be presented. In conclusion, analysis of membrane and protein networks identifies key properties of spatial organization versus pathological remodeling. The latter may directly contribute to Ca2+ release heterogeneity as a cause of pathological cellular signaling in heart failure and arrhythmias.