MCDPTA13 Participant Title Abstract

MCDPTA13

Participant Title Abstract

John Walter Acevedo Valencia Data assimilation of tree-ring- In recent years the assimilation of proxy data into climate models has appeared as a very promising procedure for reconstructing width-like observations using paleoclimate, as the states obtained in this way are in principle consistent both with historic records and the physics of the climate ensemble Kalman filtering system as represented by the model equations. In practice, however, the development of this process-based climate reconstructions techniques has been considerably hindered by several stumbling blocks, notably (i) the necessity of realistic yet affordable observation forward models able to simulate the multivariate and non-linear character of proxy response to climate forcing and (ii) the time-averaged nature of proxy information. Regarding tree-ring proxies, in particular, a new process-based forward model named VS-Lite has been shown to skilfully simulate tree-ring width chronologies coming from very different climate regimes, based solely on 3 external factors - temperature, soil moisture and solar radiation - and 5 tunable parameters. In order to investigate the applicability of VS-Lite forward model as an observation operator for tree-ring-width data within an ensemble Kalman filtering framework, we perform a set of identical twin experiments on the 2-scale Lorenz 96 model using a VS-Lite-like observation operator, which exhibits 3 special features: (i) alternating recording of two variables, (ii) saturation beyond upper and lower variable thresholds and (iii) time-averaging. As a consequence of the non-linearities of this forward model, the parameter space area where the filter has skill is substantially reduced as compared to the performance when a completely linear operator is used. We considerably counteract this undesired effect via a modification of the forward model which smooths the shifting of recorded variable.

Mohammad Ayaz Ahmad Chaotic behavior of An analysis has been made to study the chaotic behavior in the nuclear fragmentation process of 28Si–Emusion collisions at 14.5 A multiparticle production in GeV by using new parameters, erraticity spectrum and the entropy index, µq. The present study provides some evidences of the erratic relativistic heavy Ion behavior of the target fragments, suggesting chaotic target fragmentation and also spells out for the chaos in the relativistic and ultra collisions. relativistic heavy-ion collisions. Finally, it is also observed that the target fragmentation process becomes less chaotic with the raise of average multiplicities in the multiparticle production of the final states.

Afshin Akhshani Detection of degree of non- periodicity in chaotic signals based on scale index analysis

Marcus Beims Characterization of stickiness In area-preserving maps the stickiness generically occurs to the border of 1D tori so in high-dimensional that the recurrence time is a measure of the time Hamiltonian systems the trajectory spends around such structure (sticky motion) before returning to the chaotic sea. In higher dimensions, however, a long recurrence times could be originated due to tori of different dimensionalities. In this work we introduce a methodology which combines recurrence times and the number of local Lyapunov exponents close to zero, and allows for an improved characterization of stickiness in high-dimensional Hamiltonian systems. We test this procedure in a chain of coupled Standard maps and confirm that stickiness events of different recurrence time length are dominated by trajectories with different number of Lyapunov exponents.

Ezequiel Bianco-Martinez Determining functional and The study of the way information is exchanged between nodes in complex networks is of primordial importance for the understanding of physical connectivity in its functionality. A quantity known as Mutual Information Rate (MIR),defined as the mutual information per unit of time, can quantify the complex time-series exchange of information between nodes in a pair-wise fashion or between pairs of data sets, and is well defined for systems with correlation as well as memoryless systems. Our work utilizes the MIR and bounds for it to understand the relationship between information flow, functional and physical topology of complex dynamical networks. The bounds for the MIR in a network are defined in terms of dynamical invariants such as Lyapunov exponents, expansion rates, dimensions, which consequently allows one to relate information with dynamical characteristics of this network, without needing to calculate probabilities, a challenge in time-series analysis. These invariants, easy to be calculated when the dynamical equations of the network are known, require huge efforts to be calculated in real data coming from complex systems. This work shows that using simple strategies, finite resolution partitions that approximate the Markov generating partitions (whose probability densities of points belonging to them behave as if they had been generated by a random system) can be obtained. These dynamical invariants can be easily calculated from this approximate Markov partition, which leads to the estimation of the bounds for the MIR. These Markov approximations can also be used to calculate the MIR, a quantity that is different from Shannon’s entropy and mutual information, is well defined in systems with memory. This methodology is then used to determine functional and physical connections in data coming from complex dynamical networks.

Jorge C. Leitão Monte carlo sampling in open We consider the problem of detection and characterization of high-dimensional open dynamical systems (e.g., chaotic scattering). chaotic systems Chaos in these systems happens during the transient time before trajectories leave the system. The numerical difficulty in characterizing such dynamics is that the relative number of long-living trajectories decays exponentially in time and they are fractally distributed in the phase space. In this poster we introduce an efficient Monte Carlo method to sample the phase space of high-dimensional open chaotic systems which changes the scaling of the computational efficiency from exponential to polynomial as a function of the maximum escape time.

Ref.: J. C. Leitão, J. M. V. P. Lopes, E. G. Altmann, ArXiv: 1302.4672

Jens Christian Claussen Predictive control in poincare- The Ott-Grebogi-Yorke method of controlling chaos is applicable only up to a certain Ljapunov exponent if control is based on delayed based chaos control measurements [1]. A predictive control stategy can be applied straightforwardly [2]. Predictive methods however are not immediately applicable to Bielawski-Derozier-Glorieux control, or difference control, which is a Poincare-based counterpart of Pyragas control. However, within a one-dimensional family of control schemes always an optimal control scheme can be defined [2,3]. A time-continuous Floquet analys allows to investigate these schemes with respect to control impulse length [4]. In all these control schemes, predictability is delimited by the inverse of the Ljapunov number, unless memory terms are added [2,3].

[1] PRE 70, 046205 (2004) [2] PRE 70, 056225 (2004) [3] PRE 58, 7256 (1998) [4] NJP 10, 063006 (2008)

Syamal Dana Extreme Multistability in An extreme kind of multistability, where the number of coexisting attractors is infinite, has been reported in a coupled system by Sun et coupled dynamcial systems al. [1]. A particular choice of the coupling plays the key role. The coupling needed to obtain this extreme kind of multistability is rather unusual. It has been shown that the reason for the emergence of infinitely many attractors lies in the appearance of a conserved quantity in the long-term limit [1, 2]. In particular cases, this conserved quantity can be considered as a unique emergent parameter which governs the dynamics of the system in the long-term limit and, hence, allows for a reduction of the dimension of the system. However, in these studies of extreme multistability in coupled systems, it was not possible to find a systematic definition of the coupling type. In example systems, such as the Lorenz system or the coupled chemical oscillators, the coupling was particularly chosen to create an infinity of attractors. The question arises if there is a general principle of designing the coupling, leading to infinitely many attractors in coupled systems. In this presentation, we address the issue of finding a general principle of defining the coupling for extreme multistability. The coupling is defined [3] in a systematic way by using the principle of partial synchronization based on the Lyapunov function stability. The method is very general in its applicability to any dynamical system and allows many alternative design options. As a result, it provides flexibility in the physical realization of extreme multistability in dynamical systems. This phenomenon is also found to be robust with respect to parameter mismatch. The result is recently extended to network of oscillators.

[1] H. Sun, S. K. Scott, and K. Showalter, Phys. Rev. E 60, 3876(1999). [2] C. N. Ngonghala, U. Feudel, and K. Showalter, Phys. Rev. E 83, 056206 (2011). [3] C.R.Hens, U.Feudel, S.K.Dana, Phys.Rev.E 85, ,035202 (R) (2012).

Ioannis Gkolias Chaos and localization in 1- There is an ongoing research on the behavior of 1-dimensional disordered nonlinear lattices. Various results about the diffusion of an dimensional disordered initially localised wavepacket have already been published. In the present work we follow a quantitative approach to the problem. The nonlinear lattices Klein-Gordon quartic disordered lattice is used as a reference model and its numerical integration is done with the help of the innovative tangent map method. The computation of the maximal Lyapunov indicator, for the first time in this kind of lattices, give us a new insight into the problem. Various different cases of

disorder strength and initial wavepacket size are explored. Also the behavior of the deviation vector in all of the above cases is discussed.

Andy Hammerlindl The detection and analysis of slow-fast systems using Ulam's method

Nikos Kyriakopoulos Dynamical behavior of a N. Kyriakopoulos, V. Koukouloyiannis, Ch. Skokos, P. Kevrekidis, G. Voyatzis system of interacting vortices in a confined Bose-Einstein We study the dynamical behavior of a system of interacting vortices in a confined Bose-Einstein condensate. First, we consider a condensate system of 3 non-corotating vortices. In this case, we can use Poincare sections in order to distinguish the regions of regular and chaotic motion, because there exist two integrals of motion, the energy, H, and the angular momentum, L, so we can treat the latter as a parameter and treat the system as having two degrees of freedom. This is not the case when we consider a system of 4 corotating vortices. In this case the Poincare section cannot be used to get a clear image of the underlying dynamics, so we resort to the stability maps acquired by using the SALI/GALI chaotic indices.

Steffen Lange Trapping of chaotic orbits in Generic Hamiltonian systems with more than two degrees of freedom lead 4D maps to chaotic zones in phase space which are all interconnected by the Arnol'd web. We study 4D maps with a regular region embedded in a large chaotic sea, i.e. far away from the near-integrable regime. Chaotic orbits show a power-law decay of survival times. We find that the underlying mechanism is clearly different from trapping in 2D maps. Moreover, it is not related to the Arnol'd web. Instead, an anisotropic diffusion near the surface of the regular region is observed.

Xinhe Liu Neural approximations to self- Recently a new class of dynamical systems was introduced, in which the velocity vector field was shaped by the externally applied shaping vector fields signals, called self-shaping dynamical systems (SSDS) [1]. Such systems can describe the essential functions and features of the human learning, i.e. memorisation, categorisation, pattern recognition and forgetting, all occurring at the same time and automatically. They provide an alternative to a “neural network” (NN) description of learning, are potentially more powerful than, and lacking the known limitations of, the standard artifical NNs. We suggest that the performance of artificial NNs could be improved if their learning style matches that of SSDS. We attempt to represent the state and the vector field of a SSDS by a NN, and to reveal the way the inter- neuron connections could change in time to imitate the learning style of SSDS.

[1] N.B. Janson & C.J. Marsden, Self-shaping dynamical systems and learning, arXiv:1111.4443.

Clemens Löbner Integrable approximation of Our aim is to approximate the dynamics of a regular island in a non-integrable Hamiltonian H by an integrable Hamiltonian H_reg. We regular Islands: The iterative present a new method which allows to find H_reg for arbitrarily many degrees of freedom. The method is based on the construction of canonical transformation an integrable approximation in action representation using frequency analysis. The integrable approximation is improved in phase- method space representation by iterative applications of canonical transformations. These transformations are optimized such that the regular dynamics of H and H_reg agree as closely as possible.

We apply this iterative canonical transformation method to the standard map and the cosine billiard. In the second case the resulting integrable Hamiltonian describes a billiard with the same boundary, but a nontrivial time evolution. This provides a basis for the future determination of regular-to-chaotic tunneling rates for generic billiards with the fictitious integrable system approach.

Nicolas Maffione The LP-VIcode v1.0 (Kaos): A Since the early work of Hénon and Heiles (1964), the development of indicators of chaos has grown exponentially fast. However, their chaos indicator suite use is usually restricted to a few of them. Our main goal in the project is to contribute with the diversity factor. Therefore, we introduce the LP-VIcode v1.0 (Kaos), a fully operational suite of variational indicators. Written in standard Fortran and independent of the integration algorithm, the LP-VIcode v1.0 can reliably compute a group of chaos indicators from a growing library. Nowadays, the library consists of the following methods of chaos detection: the Lyapunov Characteristic Numbers, the Dynamical Spectra of Stretching Numbers and the Spectral Distance, the Fast Lyapunov Indicator and the Othogonal Fast Lyapunov Indicator, the Relative Lyapunov Indicator, the Smaller Alignment Index and the Generalized Alignment Index as well as the Mean Exponential Growth factor of Nearby Orbits.

Emad Mahmoud Projective synchronization in In this paper, the passivity theory is used to achieve projective synchronization (PS) in coupled partially linear hyperchaotic complex coupled hyperchaotic complex systems. By using this theory, the control law is thus adopted to make state vectors asymptotically synchronized up to a desired scaling systems using passive theory factor. The numerical simulations of the new hyperchaotic complex Lorenz system, as an example, are illustrated to verify the and its application to secure theoretical results. The theoretical foundation of the PS based on the passivity theory is exploited for application to secure communications communications. The numerical simulations of secure communication is used to send large message, an image and sound (voice) signal. The errors are controlled to zero which show the agreement between theoretical and numerical simulations results.

Dorothea Manika Computing the spectrum of Based on the compound matrix theory we implement appropriate differential equations for describing the time evolution of volume lyapunov exponents by elements around the orbits of dynamical systems, instead of defining these volumes through deviation vectors, whose evolution is compound matrices governed by the usual variational equations. This approach allows the accurate evaluation of Lyapunov Exponents. Using symplectic integration algorithms we apply this technique to low dimensional autonomous Hamiltonian systems, and compare its effectiveness to other known methods for the computation of Lyapunov Exponents.

Thanos Manos Dynamical localization in kicked rotator as a paradigm of other systems: spectral statistics and the localization measure

Lydia Novozhiloval Stability of invariant sets of a Equilibrium points and periodic solutions of a system of weakly coupled Landau-Lifshitz equations with spin-transfer torque are found. system of weakly coupled Rigorous analysis of stability of these invariant sets is presented. The system generalizes Landau-Lifshits model with spin transfer Landau- Lifshits equations torque that is used for modeling magnetization dynamics of an active layer in bi-layered nanodevices to the case when both layers are with spin-transfer torque active. Since the coupled system is four-dimensional, chaos is not precluded as for commonly used single Landau-Lifshitz equation. terms Some preliminary numerical results on chaos detection are presented.

Przemysaw Perlikowski Rare attractors and their We consider two systems where the rare attractors exist: externally excited van der Pol - predictability in mechanical Duffng oscillator and two mechanical clocks hanged on the same beam (Huygens experiment). systems We discuss the mechanism leading to the multistability and coexistence of multiple attractors. For van der Pol - Duffng oscillator it has been shown that the mechanism (the sequence of bifurcations) leading to the phase multistability in coupled oscillators is the same as the mechanism leading to the bistability in the single oscillator. While for clocks the multistability is caused by discontinuity of forcing momentum (escapement mechanism of clock). Obtained solutions are very sensitive for system parameters and initial conditions.

Woulache Rosalie Threshold field in nucleation We investigate numerically the effect of local inhomogeneity on the process for an nucleation process of kink-antikink pairs in the driven nonlinear Klein- inhomogeneous deformable Gordon model with the Remoissenet-Peyrard substrate potential, whose nonlinear Klein-Gordon shape can be varied as a function of the shape parameter and which has system the sine-Gordon shape as a particular case. From a numerical computation of the system under defined conditions, the configuration of the critical nucleus in the presence of a localized inhomogeneity is determined and shown to be affected by the shape parameter of the substrate potential. The expression of the depinning threshold field of kink-antikink pairs, which is the value of the applied field at which the process of the nucleation of kink-antikink pairs takes place, is also obtained. The dependence of this depinning threshold field on the shape parameter r shows that it strongly increases, for small intensity of the impurity potential when the shape deviate from the sinusoidal one.

Nicolas Rubido Inferring network structure The relationship between structure and behaviour among interacting elements forming a network is one of the most studied topics in the from non-linear method theory of complex networks. Structure is a topological representation of the interactions and it is described by the theory of graphs. measurements Behaviour is a functional observable of the network and can be measured by a variety of different methodologies. The importance of understanding this relationship is because, in nature and society, both quantities are usually not simultaneously known. For example, in the brain, behaviour can be measured but the structure is

not known. On the other hand, the structure of the current power-grid is known, but behaviour is not (e.g., black-outs and power drops).

This work aims to understand this relationship by inferring from measurements of the behaviour of a network of coupled maps the underlying structure, namely, extracting which maps are connected to which by looking at their correlated dynamics. In order to be able to contrast the findings, the structure is a known variable. The use of maps is justified due to their handling simplicity and broad spectrum of dynamical behaviours, ranging from periodic or chaotic to stochastic. Measurements are based on state-of-the-art nonlinear measurements (Reshef, Science 334, 2011, M. S. Baptista, PlosOne 7(10), 2012, Barrett, PlosOne 7(1), 2012).

Matteo Sala Tangent structures induced by We review our recent results on the connections between generators of local dilations of flows phase-space, the finite-time Lyapunov spectrum and the Oseledets' Splitting associated to any multi-dimensional flow.

Jan Schumann-Bischoff Nonlinear system An optimization based state and parameter estimation (for ordinary and delay differential equations, ODEs and DDEs) method is identification employing presented where the required sparse Jacobian matrix of the cost function is computed via automatic differentiation (AD). Automatic automatic differentiation differentiation evaluates the source code of the cost function and provides exact values of the derivatives.

Patrick Shipman Transition to chaos in gas-to- Periodic precipitation is the name given to spatially defined bands of solid precipitate produced in numerous chemical systems such as particle periodic precipitation agate rocks and concentric layers in gallstones. Here we report on a variety of periodic precipitation patterns produced in gas-to- systems particle reactions such as the reaction of ammonia (NH3) with hydrochloric acid (HCl). We examine in detail an experimental system in which NH3 and HCl are counterdiffused in a tube and find a period doubling to chaos in both the experimental system and a mathematical model. The end result of such an experiment is a series of precipitation rings left on the tube. Using tools from persistent topology who show how to extract a signature from these rings that characterizes the form of the terms governing nucleation and growth in a mathematical model.

Arkadiusz Syta Analysis of chaotic non- Shape memory materials exhibit strong thermomechanical coupling, so that temperature variations occur during mechanical loading and isothermal solutions of unloading. Instead of the standard Lyapunov exponent treatment, the statistical 0-1 test based on the asymptotic properties of a thermomechanical shape Brownian motion chain is applied to fully non-isothermal system. To improve its reliability the test memory oscillators has been applied on the time-histories of maxima and minima of each trajectory, in each component. The obtained results have been validated and confirmed by the corresponding Fourier spectra. Non-regular solutions with different levels of chaoticity have been analysed and their qualitative difference is reflected by the different values to which the control parameter $K$ asymptotically converge.

Assen Tchorbadjieff Risk models for cosmic rays After the prime cosmic ray particles penetrate the Earth atmosphere, they give birth of cascades of secondaries. The number and particle flux direction of these secondary particles are random variables in stochastic process of birth, decay and particles interactions. This process can be modeled with branching stochastic processes and any particle flux as compound counting probability distributions due to probability of survival. These probabilities can be estimated with Risk Models and computed with Generalized Linear Model for usage of real data. Moreover, the observed variances of residuals and scale parameters are compared with particle flux trend and observation of vertical atmosphere variations.

Xiaozhu Zhang Statistics, predictability and Critical transitions in multistable systems have been discussed as models for a variety of phenomena ranging from the extinctions of dynamics of critical transitions species to socio-economic changes and climate transitions between ice-ages and warm-ages. From Bifurcation theory we can expect a critical transition to be announced by a decreased recovery from external perturbations. The consequences of this critical slowing down have been observed as an increase in variance and correlation before to the transition happens. However, it is not clear, whether these changes in observation variables are statistically relevant such that they could be used as predictors for critical transitions. In this contribution we investigate the predictability of critical transitions in the Van der Pol Oscillator under the influence of external noise. We focus especially on the statical analysis of the success of predictions and the overall predictability of the system.

Yaroslav Zolotaryuk Chaotic depinning in the Kink dynamics in the periodically driven and damped disordered 1D Frenkel-Kontorova chain is investigated. The mechanism of the disordered driven Frenkel- kink depinning (locking/unlocking) transition is studied. We show that the depinning process takes place through chaotization of the Kontorova model initially stable standing mode-locked kink state. Depending on the driving parameters the chaotization process can take place through the different bifurcation scenarios. With the help of Floquet analysis the mechanism of the depinning suppression due to spatial disorder is explained.