Nonlinear Dynamics and Chaos

In time series analysis, it has been considered of key importance to determine whether a complex time series measured from the system is regular, deterministically chaotic, or random. Recently, Gottwald and Melbourne have proposed an interesting test for chaos in deterministic systems. Their analyses suggest that the test may be universally applicable to any deterministic dynamical system. In order to fruitfully apply their test to complex experimental data, it is important to understand the mechanism for the test to work, and how it behaves when it is employed to analyze various types of data, including those not from clean deterministic systems. We find that the essence of their test can be described as to first constructing a random walklike process from the data, then examining how the variance of the random walk scales with time. By applying the test to three sets of data, corresponding to ͑i͒ 1/ f ␣ noise with long-range correlations, ͑ii͒ edge of chaos, and ͑iii͒ weak chaos, we show that the test mis-classifies ͑i͒ both deterministic and weakly stochastic edge of chaos and weak chaos as regular motions, and ͑ii͒ strongly stochastic edge of chaos and weak chaos, as well as 1 / f ␣ noise as deterministic chaos. Our results suggest that, while the test may be effective to discriminate regular motion from fully developed deterministic chaos, it is not useful for exploratory purposes, especially for the analysis of experimental data with little a priori knowledge. A few speculative comments on the future of multiscale nonlinear time series analysis are made.

Various differentiable models are frequently used to describe the dynamics of complex systems (kinetic models, fluid models). Given the complexity of all the phenomena involved in the dynamics of such systems, it is required to introduce the dynamic variables dependences both on the space-time coordinates and on the resolution scales. Therefore, in this case an adequate theoretical approach may be the use of non-linear physical models. In such framework, using a simplified version of the fractal hydrodynamic model, the dynamics of a free Gaussian "perturbation" is analyzed. Possible implications of the model in dynamics of biological structures are also studied.

Using an integrated colliding-pulse mode-locked semiconductor laser, we demonstrate the existence of nonlinear dynamics and chaos in photonic integrated circuits (PICs) by demonstrating a period-doubling transition into chaos. Unlike their stand-alone counterparts, the dynamics of PICs are more stable over the lifetime of the system, reproducible from batch to batch and on faster time scales due to the small sizes of PICs.

We study the survival probability time distribution (SPTD) in dielectric cavities. In a circular dielectric cavity the SPTD has an algebraic long time behavior, ∼ t -2 in both TM and TE cases, but shows different short time behaviors due to the existence of the Brewster angle in TE case where the short time behavior is exponential. The SPTD for a stadium-shaped cavity decays exponentially, and the exponent shows a relation of γ ∼ n -2 , n is the refractive index, and the proportional coefficient is obtained from a simple model of the steady probability distribution. We also discuss about the SPTD for a quadrupolar deformed cavity and show that the long time behavior can be algebraic or exponential depending on the location of islands.

Duffing oscillator with delayed feedback is widely used in engineering. Chaos in such system plays an important role in the dynamic response of the system, which may lead to the collapse of the system. Therefore, it is necessary and significant to study the chaotic dynamical behaviors of such systems. Chaotic dynamics of the Duffing oscillator subjected to periodic external and nonlinear parameter excitations with delayed feedback are investigated both analytically and numerically in this paper. With the Melnikov method, the critical value of chaos arising from heteroclinic intersection is derived analytically. The feature of the critical curves separating chaotic and nonchaotic regions on the excitation frequency and the time delay is investigated analytically in detail. Under the corresponding system parameters, the monotonicity of the critical value to the excitation frequency, displacement time delay, and velocity time delay is obtained rigorously. The chaos threshold obtained by the analytical method is verified by numerical simulations.

We studied a magnetic turbulence axisymmetric around the unperturbed magnetic field for cases having different ratios l ʈ /l Ќ . We find, in addition to the fact that a higher fluctuation level ␦B/B 0 makes the system more stochastic, that by increasing the ratio l ʈ /l Ќ at fixed ␦B/B 0 , the stochasticity increases. It appears that the different transport regimes can be organized in terms of the Kubo number Rϭ(␦B/B 0 )(l ʈ /l Ќ ). The simulation results are compared with the two analytical limits, that is the percolative limit and the quasilinear limit. When RӶ1 weak chaos, closed magnetic surfaces, and anomalous transport regimes are found. When RϷ1 the diffusion regime is Gaussian, and the quasilinear scaling of the diffusion coefficient D Ќ ϳ(␦B/B 0 ) 2 is recovered. Finally, for Rӷ1 the percolation scaling of the diffusion coefficient D Ќ ϳ(␦B/B 0 ) 0.7 is obtained.

Atrial fibrillation ͑AF͒ is a common cardiac arrhythmia, but its mechanisms are incompletely understood. The identification of phase singularities ͑PSs͒ has been used to define spiral waves involved in maintaining the arrhythmia, as well as daughter wavelets. In the past, PSs have often been identified manually. Automated PS detection algorithms have been described previously, but when we attempted to apply a previously developed algorithm we experienced problems with false positives that made the results difficult to use directly. We therefore developed a tool for PS identification that uses multiple strategies incorporating both image analysis and mathematical convolution for automated detection with optimized sensitivity and specificity, followed by manual verification. The tool was then applied to analyze PS behavior in simulations of AF maintained in the presence of spatially distributed acetylcholine effects in cell grids of varying size. These analyses indicated that in almost all cases, a single PS lasted throughout the simulation, corresponding to the central-core tip of a single spiral wave that maintained AF. The sustained PS always localized to an area of low acetylcholine concentration. When the grid became very small and no area of low acetylcholine concentration was surrounded by zones of higher concentration, AF could not be sustained. The behavior of PSs and the mechanisms of AF were qualitatively constant over an 11.1-fold range of atrial grid size, suggesting that the classical emphasis on tissue size as a primary determinant of fibrillatory behavior may be overstated.

Order parameter equations, such as the complex Swift-Hohenberg (CSH) equation, offer a simplified and universal description that hold close to an instability threshold. The universality of the description refers to the fact that the same kind of instability produces the same order parameter equation. In the case of lasers, the instability usually corresponds to the emitting threshold, and the CSH equation can be obtained from the Maxwell-Bloch (MB) equations for a class C laser with small detuning. In this paper we numerically check the validity of the CSH equation as an approximation of the MB equations, taking into account that its terms are of different asymptotic order, and that, despite of having been systematically overlooked in the literature, this fact is essential in order to correctly capture the weakly nonlinear dynamics of the MB. The approximate distance to threshold range for which the CSH equation holds is also estimated.

This paper extends the two-compartment granular fountain ͓D. van der Meer, P. Reimann, K. van der Weele, and D. Lohse, Phys. Rev. Lett. 92, 184301 ͑2004͔͒ to an arbitrary number of compartments: The tendency of a granular gas to form clusters is exploited to generate spontaneous convective currents, with particles going down in the well-filled compartments and going up in the diluted ones. We focus upon the bifurcation diagram of the general K-compartment system, which is constructed using a dynamical flux model and which proves to agree quantitatively with results from molecular dynamics simulations.

We briefly discuss the status of the intermittency hypothesis, according to which the grand minima type variability in solar-type stars may be understood in terms of dynamical intermittency. We review concrete examples which establish this hypothesis in the mean-field setting. We discuss some difficulties and open problems regarding the establishment of this hypothesis in more realistic settings as well as its operationally decidability.

The hysteretic nonlinear dependence of pre-sliding friction force on displacement is modeled using different physics-based and black-box approaches including various Maxwell-slip models, NARX models, neural networks, nonparametric (local) models and dynamical networks. The efficiency and accuracy of these identification methods is compared for an experimental time series where the observed friction force is predicted from the measured displacement. All models, although varying in their degree of accuracy, show good prediction capability of pre-sliding friction. Finally, we show that even better results can be achieved by using an ensemble of the best models for prediction.

A model of a partially deformable Euler disk is presented that allows transverse vibrations to be treated with the techniques of classical analytical mechanics. The model clearly shows that the increasing audible frequency produced during motion can be directly related to the forcing effect of the reaction and the angular velocity of the contact point. The material of the disk seems to play a role in affecting the intensity and quality of the sound, but not its pitch. Moreover, the friction force grows rapidly with the decline of the disk, thus causing the slipping that is partially responsible for the abrupt end of the motion. The model also supports the conjecture ͓P. Kessler and O. M. O'Reilly, Regul. Chaotic Dyn. 7, 49 ͑2002͔͒ that the vibrations themselves contribute to this phenomenon by causing a loss of contact with the surface at small angles of inclination.

This colloquium gives an overview of recent theoretical and experimental progress in the area of nonequilibrium dynamics of isolated quantum systems. We particularly focus on quantum quenches: the temporal evolution following a sudden or slow change of the coupling constants of the system Hamiltonian. We discuss several aspects of the slow dynamics in driven systems and emphasize the universality of such dynamics in gapless systems with specific focus on dynamics near continuous quantum phase transitions. We also review recent progress on understanding thermalization in closed systems through the eigenstate thermalization hypothesis and discuss relaxation in integrable systems. Finally we overview key experiments probing quantum dynamics in cold atom systems and put them in the context of our current theoretical understanding. I. Introduction 1 II. Nearly adiabatic dynamics in quantum systems 3 A. Universality in a nutshell 3 B. Universal dynamics near quantum critical points 3 C. Slow dynamics in gapped and gapless systems. 7 D. Effects of finite temperature. 8 E. Open problems 9 III. Effects of Integrability and its breaking: ergodicity and thermalization. 9 A. Quantum and classical ergodicity. 10 B. Nonergodic behavior in integrable systems: the generalized Gibbs ensemble 11 C. Breaking integrability: eigenstate thermalization. 14 D. Outlook and open problems: quantum KAM threshold as a many-body delocalization transition ? 15 IV. Experimental progress in quantum dynamics in cold atoms and other systems 16 V. Outlook 20 VI. acknowledgements 20 References 20

Understanding the behavior of the Logistic Equation is critical for those giving their first steps in the study of Chaos theory and, the best way to get familiar with chaotic dynamics is by developing computer programs to simulate it, with this in mind, this article deals mainly with the computational algorithms to study the dynamics of the mentioned equation. Implementing the computer programs with the algorithms in this work allow visualizing the dependence on initial conditions of the logistic equation, a feature typical of chaotic systems, this is made evident by running the logistic equation starting from slightly different initial conditions and may be graphically shown under slow motion on computer screen. A special case of dependence on initial conditions is the Butterfly effect, which may also be visualized with the algorithms included in this chapter. The reported algorithms also allow experimentally assessing up to five values of Feigenbaum's delta from the plotting of the bifurcations cascade of the equation. This chapter deals also with the parabolic shaped Maps of Return of the logistic equation.

Cells in multicellular organisms switch between distinct cell fates, such as proliferation or differentiation into specialized cell types. Genome-wide gene regulatory networks govern this behavior. Theoretical studies of complex networks suggest that they can exhibit ordered (stable) dynamics, raising the possibility that cell fates may represent high-dimensional attractor states. We used gene expression profiling to show that trajectories of neutrophil differentiation converge to a common state from different directions of a 2773-dimensional gene expression state space, providing the first experimental evidence for a high-dimensional stable attractor that represents a distinct cellular phenotype.

An accurate modeling of a wavy film flow down an inclined plane is developed using the weighted residual technique which was first proposed by Ruyer-Quil and Manneville [Eur. Phys. J. B 15 (2000) 357]. The model includes third order terms in order to better capture the effects of small Weber and high Reynolds numbers. This is made possible by an appropriate refinement of the velocity profile. To this end, a free parameter α acting on the flexibility of the velocity profile is introduced. It is shown, from linear stability analysis that the model follows quite closely, for a suitable choice of α, the Orr-Sommerfeld equation for all Weber and Kapitza numbers. The improvement is of course more substantial in the inertia dominated regimes. Some prominent qualitative and quantitative characteristics of traveling wave solutions are then derived from a simplified version of the model that is before hand converted into a three dimensional dynamical system.

A generic feature of systems with long-range interactions is the presence of quasistationary states with non-Gaussian single particle velocity distributions. For the case of the Hamiltonian mean-field model, we demonstrate that a maximum entropy principle applied to the associated Vlasov equation explains known features of such states for a wide range of initial conditions. We are able to reproduce velocity distribution functions with an analytic expression which is derived from the theory with no adjustable parameters. A normal diffusion of angles is detected, which is consistent with Gaussian tails of velocity distributions. A dynamical effect, two oscillating clusters surrounded by a halo, is also found and theoretically justified.

In this paper, we study the relationship between the phase transition and Lyapunov exponents for 4D Hayward anti-de Sitter (AdS) black hole. We consider the motion of massless and massive particles around an unstable circular orbit of the Hayward AdS black hole in the equatorial plane and calculate the corresponding Lyapunov exponents. The phase transition is found to be well described by the multivaled Lyapunov exponents. It is also found that different phases of Hayward AdS black hole coincide with different branches of the Lyapunov exponents. We also study the discontinuous change in the Lyapunov exponents and find that it can serve as an order parameter near the critical point. The critical exponent of change in Lyapunov exponent near the critical point is found to be 1/2.

A trimer is an object composed of three centimetrical stainless steel beads equidistant from each other and is predestined to show richer behaviour than a bouncing ball or a bouncing dimer. A rigid trimer was placed on the plate of an electromagnetic shaker and was vertically vibrated according to a sinusoidal signal. The horizontal translational and rotational motions of the trimer were recorded for a range of frequencies between 25 and 100 Hz, while the amplitude of the forcing vibration was tuned to obtain maximal acceleration of the plate up to 10 times gravity. Several modes have been detected such as, e.g., rotational and pure translational motions. These modes are found at determined accelerations of the plate and do not depend on the frequency. Chaotic behaviour is observed for other accelerations. By recording the time delays between two successive contacts when the frequency and the amplitude are fixed, a map of the bouncing regime was constructed and compared with that of the dimer and the bouncing ball. Period-2 and period-3 orbits were experimentally observed. In these modes, according to observations, the contact between the trimer and the plate is persistent between two successive jumps. This persistence erases the memory of the jump preceding the contact. A model based on the conditions for obtaining persistent contact is proposed and allows us to explain the values of the particular accelerations for which period-2 and period-3 modes were observed. Finally, numerical simulations allow us to reproduce the experimental results. This allows us to conclude that the friction between the beads and the plate is the major dissipative process.

This article introduces a new semi-analytical nonlinear finite element formulation for thin cylinders according to a continuum-based approach. A comparison between the continuum-based approach and the classical approach for the buckling behavior of isotropic and orthotropic perfect cylinders validates the results. The classical approach is defined according to thin shell theories based on the von Karman approximation. A mathematical modeling for geometry imperfection of cylinders is derived according to a continuum-based approach whose results are compared with the results of the classical approach for imperfect cylinders. The influence of neglecting some nonlinear terms in the classical approach for perfect and imperfect cylinders on the buckling path is investigated. In the buckling analysis, two methods, i.e. the perturbation and load disturbance methods, which undertake to switch to the post-buckling path, are compared to each other.

We experimentally investigate the effects of polymer additives on the collective dynamics of swarming Serratia marcescens in quasi two-dimensional (2D) liquid films. We find that even minute amounts of polymers (≤ 20 ppm) can significantly enhance swimming speed and promote largescale coherent structures. Velocity statistics show that polymers suppress large velocity fluctuation, transforming the velocity distributions from super-Gaussian to Gaussian. Spatial and temporal correlation functions suggest that polymers increase both the size and lifetime of flow structures. The energy spectra show an exponential decay at low wavenumbers, with a characteristic length scale increasing with polymer concentration. Overall, these result show polymers can mediate bacteria interaction and promote large-scale coherence in dense active suspensions.

A brief review is made of the birth and evolution of the ''nonequilibrium potential'' (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, the NEP adds a quantitative dimension to the qualitative theory of differential equations and provides a global Lyapunov function for the deterministic dynamics. Here we illustrate the usefulness of the NEP to draw results on stochastic thermodynamics: the Jarzynski equality in the Wilson-Cowan model (a population-competition model of the neocortex) and a ''thermodynamic uncertainty relation'' (TUR) in the KPZ equation (the stochastic field theory of kinetic interface roughening). Additionally, we discuss system-size stochastic resonance in the Wilson-Cowan model and relevant aspects of KPZ phenomenology like the EW-KPZ crossover and the memory of initial conditions.

Oscillating interfaces are found in the one-dimensional complex Ginzburg-Landau equation subject to a periodic force. By introducing a suitable section in the phase space, it is shown that the oscillation starts with a Hopf bifurcation and that subsequent bifurcations lead to various motions of interface. Interfacial motion plays crucial roles in the long-time behavior of spatially extended systems with bistabiIity. Recent studies revealed the nature of interaction between interfaces,l),Z) universal scaling in the motion of random interfaces,3) and instability of interface associated with negative surface tension. 4 ) In one-dimensional systems, there is no curvature flow characteristic of higher dimensional systems. One may suspect that the interface is either stationary or steadily propagating and that complex motion like 'breathing' can only arise from the interaction between interfaces. 5

We report the long-time nonlinear dynamical evolution of ultracold atomic gases in the P-band of an optical lattice. A Bose-Einstein condensate (BEC) is fast and efficiently loaded into the Pband at zero quasi-momentum with a non-adiabatic shortcut method. For the first one and half milliseconds, these momentum states undergo oscillations due to coherent superposition of different bands, which are followed by oscillations up to 60ms of a much longer period. Our analysis shows the dephasing from the nonlinear interaction is very conducive to the long-period oscillations induced by the variable force due to the harmonic confinement.

Quasicrystals are fractal due to their self similar property. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = φ. The corresponding pointwise dimension is 0.7. Various modifications, such as truncation from the head or tail, scrambling the orders of the sequence, and changing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to changes in the Fibonacci order but not to the L/S ratio.

Symplectic integration methods based on operator splitting are well established in many branches of science. For Hamiltonian systems which split in more than two parts, symplectic methods of higher order have been studied in detail only for a few special cases. In this work, we present and compare different ways to construct high order symplectic schemes for general Hamiltonian systems that can be split in three integrable parts. We use these techniques to numerically solve the equations of motion for a simple toy model, as well as the disordered discrete nonlinear Schrödinger equation. We thereby compare the efficiency of symplectic and non-symplectic integration methods. Our results show that the new symplectic schemes are superior to the other tested methods, with respect to both long term energy conservation and computational time requirements.

We demonstrate that meandering as well as regular spiral waves can form in a well-controlled culture layer of rat ventricle cells and that the meandering spiral wave, in particular, can generate an alternant rhythm. These observations are made possible by a newly developed, noninvasive phase contrast macrooptics that is simple but highly effective in visualizing the contractile motion of the populations of cardiac cells.

Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimensional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental relevance, paradigmatically including the Michelson-Sivashinsky system. For a whole parameter range, the asymptotic dynamics is scale invariant with dimension-independent exponents reflecting a hidden Galilean symmetry. The usual Kardar-Parisi-Zhang nonlinearity, albeit irrelevant in that parameter range, plays a key role in this behavior.

Spin waves (SWs) in magnetic nanotubes have shown interesting nonreciprocal properties in their dispersion relation, group velocity, frequency linewidth, and attenuation lengths. The reported chiral effects are similar to those induced by the Dzyaloshinskii–Moriya interaction but originating from the dipole–dipole interaction. Here, we show that the isotropic-exchange interaction can also induce chiral effects in the SW transport; the so-called Berry phase of SWs. We demonstrate that with the application of magnetic fields, the nonreciprocity of the different SW modes can be tuned between the fully dipolar governed and the fully exchange governed cases, as they are directly related to the underlying equilibrium state. In the helical state, due to the combined action of the two effects, every single sign combination of the azimuthal and axial wave vectors leads to different dispersions, allowing for a very sophisticated tuning of the SW transport. A disentanglement of the dipole–dipo...

Stable domain walls which are realized by a defect between oppositely traveling spiral waves in a patternforming hydrodynamic system, i.e., Taylor-Couette flow, are studied numerically as well as experimentally. A nonlinear mode coupling resulting from the nonlinearities in the underlying momentum balance is found to be essential for the stability of the defects. These nonlinearly driven defects separate spiral domains and act either as a phase generating or annihilating defect. Specific phase differences of either 0 or between the participating traveling waves are a characteristic feature of this defect. The influence of a symmetry breaking externally imposed flow on the spiral domains and the defects is studied. The numerical and experimental results are in excellent agreement.

Stroboscopic illumination of a rapidly rotating disk with radial spokes leads to a range of different stationary and moving images as the angular rotation frequency of the disk and the strobe frequency are varied. We compare predictions from the standard correlation model of motion perception with a model based on phase locking observed during periodic stimulation of an integrate-and-fire nonlinear oscillator. The close agreement between theoretical predictions and experimental observations suggests the possibility that periodic forcing of nonlinear neural oscillations may play a role in motion perception.

The addition of a drug that specifically blocks a potassium channel in spontaneously beating aggregates of chick heart cells leads to complex bifurcations over time. A stochastic partial differential equation model based on discrete ionic currents recorded in these cells demonstrates that drug diffusion and noise can induce the coupled beats and bursting rhythms observed. These results provide further evidence that stochastic events at a subcellular level are needed to understand complex cardiac arrhythmias and play an important role in the onset of these arrhythmias.

We study numerically the wavefunction evolution of a Bose-Einstein condensate in a Bunimovich stadium billiard being governed by the Gross-Pitaevskii equation. We show that for a moderate nonlinearity, above a certain threshold, there is emergence of dynamical thermalization which leads to the Bose-Einstein probability distribution over the linear eigenmodes of the stadium. This distribution is drastically different from the energy equipartition over oscillator degrees of freedom which would lead to the ultra-violet catastrophe. We argue that this interesting phenomenon can be studied in cold atom experiments.

The European Project Solvency II is devoted to the appraisal of a Solvency Capital Requirement that should capture the overall risk profile of insurance companies. In this framework there is a growing need to develop so-called internal risk models to get accurate estimates of liabilities. In the context of non-life insurance, it is crucial to correctly assess risk from different sources, such as underwriting risk with particular reference to premium, reserving and catastrophe risks. In particular the underwriting cycle is not quantified in standard formula under Quantitative Impact Study 4, but probably it could be included, as it provides additional volatility to liabilities distribution and so it could increase the capital requirement. The aim of this paper is to correctly model the underwriting cycle for non-life insurance companies, also taking into account its effect on the solvency ratio. Starting from Collective Risk Theory, a dynamic control policy is defined to specify the relationship between solvency ratio and safety loading, to model the underwriting cycle. The corresponding dynamic equation for the solvency ratio, under some assumptions, assumes the form of a one dimensional piecewise linear map. In the first part of the work a deterministic version of this map is analyzed, where aggregate losses are simply regarded as a parameter. Numerical analysis and stochastic assessments of the model conclude the work.

Superlattice standing waves arising on the surface of ferrofluids that are driven by an ac magnetic field are investigated experimentally. Several different types are obtained through successive spatial period doublings, which are mediated by resonant mode interactions. The observed superlattices are quite diverse, depending on the relevant base Fourier modes, the orientation and the number of emerged subharmonic modes, and the phase difference among the involved modes all together. On the other hand, their temporal evolutions are all either period-1 ͑harmonic͒ or period-2 ͑subharmonic͒.

In this paper, we have made an attempt to provide a unified framework to understand the complex spatiotemporal patterns induced by self and cross diffusion in a spatial Holling-Tanner model for phytoplankton-zooplankton-fish interaction. The effect of critical wave length which can drive the system to instability is investigated. We have examined the criterion between two cross-diffusivity (constant and timevarying) on the stability of the model system and for diffusive instability to occur. Based on these conditions and by performing a series of extensive simulations, we observed the irregular patterns, stationary strips, spots, and strips-spots mixture patterns. Numerical simulation results reveal that the regular strip-spot mixture patterns prevail over the whole domain on increasing the values of self-diffusion coefficients of phytoplankton and zooplankton and the dynamics of the system do not undergo any further changes.

In this paper, we investigate a Rosenzweig-McAurthur model and its variant for phytoplankton, zooplankton and fish population dynamics with Holling type II and III functional responses. We present the theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially distributed population with local diffusion. The choice of parameter values is important to study the effect of diffusion, also it depends more on the nonlinearity of the system. With the help of numerical simulations, we observe the formation of spatiotemporal patterns both inside and outside the Turing space.

This work is a theoretical investigation of the stability of the non-linear behavior of an oscillating tip-cantilever system used in dynamic force microscopy. Stability criterions are derived that may help to a better understanding of the instabilities that may appear in the dynamic modes, Tapping and NC-AFM, when the tip is close to a surface. A variational principle allows to get the temporal dependance of the equations of motion of the oscillator as a function of the non-linear coupling term. These equations are the basis for the analysis of the stability. One find that the branch associated to frequencies larger than the resonance is always stable whereas the branch associated to frequencies smaller than the resonance exhibits two stable domains and one unstable. This feature allows to re-interpret the instabilities appearing in Tapping mode and may help to understand the reason why the NC-AFM mode is stable.

In this study, we examine the nonlinear interdependence between the parameters of the solar wind influencing geomagnetic activity (proton density and solar wind dynamic pressure, IMF B z and solar wind dynamic pressure) and geomagnetic indices (AE and SYM-H) during pre-storm, storm and post-storm of intense, major, moderate and minor geomagnetic storms. Nonlinear analytical tools comprising Cross Recurrence Plot (CRP) and Recurrence Rate (RR) was used to capture the features of nonlinear interdependence in the time series data of the solar wind parameters (solar wind dynamic pressure, IMF B z and Proton density) and geomagnetic indices (AE and SYM-H). The Pearson correlation coefficient was also used to reveal the features of linear dependence between the parameters of the solar wind during the different categories of geomagnetic storms. Our result shows that during the storm, the CRPs of the solar wind parameters investigated pictured a strong deterministic structure of CRP which is an indication of strong interdependence. In particular, the results of our analysis showed that the solar wind dynamic pressure and proton density depicted very strong interdependence features. On the other hand, during pre-storm and post-storm, the CRP of solar wind dynamic pressure in relation to IMF B z and proton density pictured a rare deterministic structure signifying weak interdependence. Furthermore, the values of RR were very high (an indication of strong interdependence between the parameters of the solar wind) during the storm while at pre-storm and post-storm, the values of RR declined significantly in most of the periods which further strengthened the evidence of weak interdependence in the parameters of solar wind. The CRP and RR of the geomagnetic indices (AE and SYM-H) reveals no significant difference between pre-storm, storm and post-storm of the different categories of geomagnetic storms. However, strong interdependence between the geomagnetic indices were observed in most of the periods investigated. Finally, the Pearson correlation coefficient reveals high values of linear dependence between the solar wind parameters without any significant difference between pre-storm, storm and post-storm periods during intense, major, moderate and minor geomagnetic storms.

We show that work done by the non-conservative forces along a stable limit-cycle attractor of a dissipative dynamical system is always equal to zero. Thus, mechanical energy is preserved on average along periodic orbits. This balance between energy gain and energy loss along different phases of the self-sustained oscillation is responsible for the existence of quantized orbits in such systems. Furthermore, we show that the instantaneous preservation of projected phase-space areas along quantized orbits describes the neutral dynamics of the phase, allowing us to derive from this equation the Wilson-Sommerfeld-like quantization condition. We apply our general results to near-Hamiltonian systems, identifying the fixed points of Krylov-Bogoliubov radial equation governing the dynamics of the limit cycles with the zeros of the Melnikov function. Moreover, we relate the instantaneous preservation of the phase-space area along the quantized orbits to the second Krylov-Bogoliubov equation describing the dynamics of the phase. We test the two quantization conditions in the context of hydrodynamic quantum analogs, where a megastable spectra of quantized orbits have recently been discovered. Specifically, we use a generalized pilot-wave model for a walking droplet confined in a harmonic potential, and find a countably infinite set of nested limit cycle attractors representing a classical analog of quantized orbits. We compute the energy spectrum and the eigenfunctions of this self-excited system.

We give a theoretical study of unusual resistive (dynamic) localized states in anisotropic Josephson junction ladders, driven by a DC current at one edge. These states comprise nonlinearly coupled rotating Josephson phases in adjacent cells, and with increasing current they are found to expand into neighboring cells by a sequence of sudden jumps. We argue that the jumps arise from instabilities in the ladder's superconducting part, and our analytic expressions for the peculiar voltage (rotational frequency) ratios and I-V curves are in very good agreement with direct numerical simulations.

We present a theoretical study of inhomogeneous dynamic (resistive) states in a single plaquette consisting of three Josephson junctions. Resonant interactions of such a breather state with electromagnetic oscillations manifest themselves by resonant current steps and voltage jumps in the current-voltage characteristics. An externally applied magnetic field leads to a variation of the relative shift between the Josephson current oscillations of two resistive junctions. By making use of the rotation wave approximation analysis and direct numerical simulations we show that this effect allows to effectively control the breather instabilities, e. g. to increase (decrease) the height of the resonant steps and to suppress the voltage jumps in the current-voltage characteristics.

We study the resonant scattering of plasmons ͑linear waves͒ by discrete breather excitations in Josephson junction ladders. We predict the existence of Fano resonances, and find them by computing the resonant vanishing of the transmission coefficient. We propose an experimental setup of detecting these resonances, and conduct numerical simulations which demonstrate the possibility to observe Fano resonances in the plasmon scattering by discrete breathers in Josephson junction ladders.

We investigate the origin of order in the low-lying spectra of many-body systems with random two-body interactions. Contrary to the common belief our study based both on analytical as well as on numerical arguments shows that these are the higher $J$-sectors whose ground states are more orderly than the ones in the J=0-sector. A predominance of J=0 ground states turns out to be the result of putting on together states with different characteristic energy scales from different $J$-sectors.