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Abstract Damping induced by frictional slipping in assembled structures is always a difficulty in vibration mechanics. Sometimes the cycling movement in riveted or bolted joint is generated by in-plane force field which is directly related to transverse deflection. Since this phenomenon cannot be described by classical plate theory, a general formulation based on von-Kármán plate theory is proposed in this study. The response estimation involving damping is conducted with the Krylov-Bogoliubov linearization method (HBM). The notion of non-linear mode is introduced in the context of single mode method and multi-mode method. The influence of mode shape change on forced response is also analyzed. [ABSTRACT FROM AUTHOR]

In this paper, decentralised fixed modes (DFMs) of a networked system are studied. The network topology is directed and weighted and the nodes are higher-dimensional linear time-invariant (LTI) dynamical systems. The effects of the network topology, the node-system dynamics, the external control inputs, and the inner interactions on the existence of DFMs for the whole networked system are investigated. A necessary and sufficient condition for networked multi-input/multi-output (MIMO) systems in a general topology to possess no DFMs is derived. For networked single-input/single-output (SISO) LTI systems in general as well as some typical topologies, some specific conditions for having no DFMs are established. It is shown that the existence of DFMs is an integrated result of the aforementioned relevant factors which cannot be decoupled into individual DFMs of the node-systems and the properties solely determined by the network topology. [ABSTRACT FROM AUTHOR]

The determination of unactuated structural joint axes within a compliant structure by external measurements is a subject of interest in many areas. Most eminently, the issue arises for example in the identification of a human joint axis by non-invasive measures, as is needed in robotics rehabilitation and surgery planning procedures. While most existing non-invasive approaches rely here on external optical tracking systems and/or inertial measurement units (IMU) attached to the skin and hence induce large artifact errors, identification of joint axes through tactile control has not yet been addressed in the literature. This paper presents a novel approach to determine an unknown unactuated joint axis in a compliant embedding serial kinematic chain using robot force-controlled motion. The approach consists in iteratively imparting a load increment according to the Krylov power iteration method until convergence to the force eigenscrew with largest impedance eigenvalue is achieved. The novelty of the approach consists in applying the Krylov steps directly on the target system, hence avoiding the need to explicitly identify the impedance matrix, which is difficult to obtain. The paper validates the proposed procedure by a mimicry mechanical device of the upper limb, composed of a revolute joint representing the elbow joint and a compliant wrist suspension imitating the local movements of the soft tissue at the wrist. The experiments show a promising perspective of the approach. [ABSTRACT FROM AUTHOR]

In this paper the iteration of soft continuous functions is investigated and their discrete dynamical systems in soft topological spaces are defined. Some basic concepts related to discrete dynamical systems (such as soft ω-limit set, soft invariant set, soft periodic point, soft nonwandering point, and soft recurrent point) are introduced into soft topological spaces. Soft topological mixing and soft topological transitivity are also studied. At last, soft topological entropy is defined and several properties of it are discussed. [ABSTRACT FROM AUTHOR]

This article studies KL*-stability (the stability expressed by KL*-class function) for a class of hybrid dynamical systems (HDS). The notions ofKL*K*-property and KL*-stability are proposed for HDS with respect to the hybrid-event-time. The KL*-stability, which is based on K or L property of the continuous flow, the discrete jump and the event in an HDS, extends theKLL-stability and the event-stability reported in the literature for HDS. The relationships between KL*K*-property and KL*-stability are established via introducing the hybrid dwell-time condition (HDT). The HDT generalizes the average dwell-time condition in the literature. For an HDS with KL*K*-property consisting of stabilizing L-property and destabilizing K-property, it is shown that there exists a common HDT under which the HDS will achieve KL*-stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion ofKL*-stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In the article will be reviewed the changes of ship motion patterns originating from a special class of phase space phenomena identified in the nonlinear dynamics literature as homoclinic (When the term appears in italic, it will imply truly homoclinic as well as heteroclinic events). The key incident is the contact of manifolds of saddle-type (i.e., unstable) objects, which triggers a global change in the phase space flow of the dynamical system. Homoclinic phenomena are more likely to appear in extreme dynamic behavior where, usually, the effect of nonlinearity is large. Their presence can be felt through the sudden termination of oscillatory motions; or, through the emergence of motion irregularity accompanied by loss of system robustness to external excitations. Homoclinic phenomena have been identified to lurk behind some well-known manifestations of ship dynamic instability. All known ship-related cases will be reviewed and explained in the current article. Analytical and numerical tools that can be used for predicting such phenomena effectively will be discussed, as well as, practical assessmentmethods that are suitable for evaluating ship vulnerability. Three types of ship motion will be reviewed: 1) rolling in beam seas; 2) surf-riding and broaching-to in following/quartering seas; and 3) drifting and yawing inwind. For the roll motions, the investigation will be escalated from free rolling with bias, to resonant (periodically forced) rolling and the mechanism of capsize in beam seas through safe basin erosion; to stochastic rolling and the consideration of wave grouping excitations. In astern seas, the most profoundhomoclinic event is the transition from ordinary surging to surf-riding. It will be explained, initially, in a deterministic context through simple dynamical analogues. Differences arising from the level of detail in the mathematical model will be discussed (many degrees of freedom, memory effect, and nonlinear waves). Then, a multifrequency wave environmentwill be introduced, rendering the phase space flow timevarying and incurring significant qualitative effects. Themonitoring of Lagrangian structures appearing in such unsteady phase space flows can offer unique insights about nonlinear ship motions and their interactions. Wave celerity is known to correlate with the occurrence of surf-riding in regular seas, although for irregular seas, its role has been uncertain. A phenomenological approach will be reviewed for evaluating this connection. Two possible methods for celerity's calculation in irregular seas will be presented. The third area where homoclinic phenomena have been identified is planar ship motions under strong wind loads. Two particular cases will be addressed: the transition from a course-keeping pattern to a turning motion pattern as the rudder angle is varied; and phenomena of "head wind" ship operation, under sluggish rudder control. The articlewill conclude with some thoughts as to the future direction in the field. [ABSTRACT FROM AUTHOR]

Generalized competitive modes (GCM) have been used as a diagnostic tool in order to analytically identify parameter regimes which may lead to chaotic trajectories in a given first order nonlinear dynamical system. The approach involves recasting the first order system as a second order nonlinear oscillator system, and then checking to see if certain conditions on the modes of these oscillators are satisfied. In the present paper, we will consider the inverse problem of GCM: If a system of second order oscillator equations satisfy the GCM conditions, can we then construct a first order dynamical system from it which admits chaotic trajectories? Solving the direct inverse problem is equivalent to finding solutions to an inhomogeneous form of the Euler equations. As there are no general solutions to this PDE system, we instead consider the problem for restricted classes of functions for autonomous systems which, upon obtaining the nonlinear oscillatory representation, we are able to extract at least two of the modes explicitly. We find that these methods often make finding chaotic regimes a much simpler task; many classes of parameter-function regimes that lead to nonchaos are excluded by the competitive mode conditions, and classical knowledge of dynamical systems then allows us to tune the free parameters or functions appropriately in order to obtain chaos. To find new hyperchaotic systems, a similar approach is used, but more effort and additional considerations are needed. These results demonstrate one method for constructing new chaotic or hyperchaotic systems. [ABSTRACT FROM AUTHOR]

We prove two general theorems that determine the Lie and the Noether point symmetries for the equations of motion of a dynamical system thatmoves in a general Riemannian space under the action of a time dependent potential W(t, x) =ω(t)V(x). We apply the theorems to the case of a time dependent central potential and the harmonic oscillator and determine all Lie and Noether point symmetries. Finally we prove that these theorems also apply to the case of a dynamical system with linear dumping and study two examples. [ABSTRACT FROM AUTHOR]

When the corresponding retinal locations in the two eyes are presented with incompatible images, a stable percept gives way to perceptual alternations in which the two images compete for perceptual dominance. As perceptual experience evolves dynamically under constant external inputs, binocular rivalry has been used for studying intrinsic cortical computations and for understanding how the brain regulates competing inputs. Converging behavioral and EEG results have shown that binocular rivalry and attention are intertwined: binocular rivalry ceases when attention is diverted away from the rivalry stimuli. In addition, the competing image in one eye suppresses the target in the other eye through a pattern of gain changes similar to those induced by attention. These results require a revision of the current computational theories of binocular rivalry, in which the role of attention is ignored. Here, we provide a computational model of binocular rivalry. In the model, competition between two images in rivalry is driven by both attentional modulation and mutual inhibition, which have distinct selectivity (feature vs. eye of origin) and dynamics (relatively slow vs. relatively fast). The proposed model explains a wide range of phenomena reported in rivalry, including the three hallmarks: (i) binocular rivalry requires attention; (ii) various perceptual states emerge when the two images are swapped between the eyes multiple times per second; (iii) the dominance duration as a function of input strength follows Levelt's propositions. With a bifurcation analysis, we identified the parameter space in which the model's behavior was consistent with experimental results. [ABSTRACT FROM AUTHOR]

Interrelation of various approaches to the definition of the concept of entropy determines its wide using in applications. We consider entropy characteristics used in various research techniques for investigation of complex dynamical systems (including symbolic ones) behaviour. Methods of analysis of dynamical systems phase portraits, which are based on Rényi entropy, fractal and multifractal characteristics, and results of numerical experiments are given. [ABSTRACT FROM AUTHOR]

We prove a theorem on the exact asymptotic relations of large deviations of the Bogoliubov measure in the L norm for p = 4, 6, 8, 10 with p > p , where p = 2+4 π / β ω is a threshold value, β > 0 is the inverse temperature, and ω > 0 is the natural frequency of the harmonic oscillator. For the study, we use the Laplace method in function spaces for Gaussian measures. [ABSTRACT FROM AUTHOR]

Mathematical model of a wind-powered vehicle is proposed. The vehicle uses a wing to convert the energy of an upcoming wind flow into the energy of the mechanical motion in such a way that the vehicle goes against the wind. The corresponding dynamical system is constructed. Parametric analysis of the system is carried out using the Poincare-Pontryagin asymptotic approach. The existence of an attracting steady regime is shown. Boundaries of applicability of asymptotic estimations are analyzed using numerical simulation. Results of the parametric analysis were used for construction of a prototype of such a vehicle. Experiments with this prototype qualitatively agree with the mathematical modeling. [ABSTRACT FROM AUTHOR]

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP -complete, even when the underlying graph of the dynamical system is a simple path. We identify a fixed parameter tractable problem in this context. We also consider constrained versions of threshold inference problems where the input includes a set of equality or inequality constraints (which specify pairs of nodes which must have the same threshold value or different threshold values). We present algorithmic and complexity results for several constrained threshold inference problems. [ABSTRACT FROM AUTHOR]

We have studied a Filippov system with small , and being periodic. Since is an abstract function, the subharmonic Melnikov function cannot be computed. In other words, for this system the Melnikov method loses effectiveness. First, we proved that the equation has a unique harmonic solution, a unique -subharmonic solution for any and they are Lyapunov asymptotically stable. Moreover, this equation has no other type of periodic solutions. Further, the attractor of this system is not chaotic. Finally, some numerical examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we apply the differential Galoisian approach to investigate the meromorphic non-integrability of a class of 3D equations in mathematical physics, including Nosé-Hoover equations, the Lü system, the Rikitake-like system and Rucklidge equations, which are well known in the fields of molecular dynamics, chaotic theory and fluid mechanics, respectively. Our main results show that all these considered systems are, in fact, non-integrable in nearly all parameters. [ABSTRACT FROM AUTHOR]

Recently, barrier certificates have been introduced to prove the safety of continuous or hybrid dynamical systems. A barrier certificate needs to exhibit some barrier function, which partitions the state space in two subsets: the safe subset in which the state can be proved to remain and the complementary subset containing some unsafe region. This approach does not require any reachability analysis, but needs the computation of a valid barrier function, which is difficult when considering general nonlinear systems and barriers. This paper presents a new approach for the construction of barrier functions for nonlinear dynamical systems. The proposed technique searches for the parameters of a parametric barrier function using interval analysis. Complex dynamics with bounded perturbations can be considered without needing any relaxation of the constraints to be satisfied by the barrier function. [ABSTRACT FROM AUTHOR]

The two main sources of internal friction in a rotor-shaft system are the shaft structural hysteresis and the possible shrink-fit release of the assembly. The internal friction tends to destabilize the over-critical rotor running, but a remedy against this effect may be provided by a proper combination of some external damping in the supports and an anisotropic arrangement of the support stiffness, or at most by the support damping alone, depending on the system geometry. The present analysis reported here considers a general asymmetric rotor-shaft system, where the rotor is perfectly rigid and is constrained by viscous–flexible supports having different stiffnesses on two orthogonal planes. The internal friction is modelled by nonlinear Coulombian forces, which counteract the translational motion of the rotor relative to a frame rotating with the shaft ends. The nonlinear equations of motion are dealt with using an averaging approach based on the Krylov-Bogoliubov method with some adaptation to address the multi-degree-of-freedom nature of the problem. Stable limit cycles may be attained by the overcritical whirling motions, whose amplitudes are inversely proportional to the external dissipation applied by the supports. A noteworthy result is that the stiffness anisotropy of the supports is recognized as beneficial in reducing the natural whirl amplitudes, albeit mainly in the symmetric configuration of the rotor at the mid span and, to a rather lesser extent, in the asymmetric configuration, which then requires a stronger damping action in the supports. [ABSTRACT FROM AUTHOR]

Motivation: Dynamics of cellular processes are often studied using mechanistic mathematical models. These models possess unknown parameters which are generally estimated from experimental data assuming normally distributed measurement noise. Outlier corruption of datasets often cannot be avoided. These outliers may distort the parameter estimates, resulting in incorrect model predictions. Robust parameter estimation methods are required which provide reliable parameter estimates in the presence of outliers. Results: In this manuscript, we propose and evaluate methods for estimating the parameters of ordinary differential equation models from outlier-corrupted data. As alternatives to the normal distribution as noise distribution, we consider the Laplace, the Huber, the Cauchy and the Student’s t distribution. We assess accuracy, robustness and computational efficiency of estimators using these different distribution assumptions. To this end, we consider artificial data of a conversion process, as well as published experimental data for Epo-induced JAK/STAT signaling. We study how well the methods can compensate and discover artificially introduced outliers. Our evaluation reveals that using alternative distributions improves the robustness of parameter estimates. [ABSTRACT FROM AUTHOR]

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails. [ABSTRACT FROM AUTHOR]

In the present paper, we describe an adaptive modified rational global Lanczos algorithm for model-order reduction problems using multipoint moment matching-based methods. The major problem of these methods is the selection of some interpolation points. We first propose a modified rational global Lanczos process and then we derive Lanczos-like equations for the global case. Next, we propose adaptive techniques for choosing the interpolation points. Second-order dynamical systems are also considered in this paper, and the adaptive modified rational global Lanczos algorithm is applied to an equivalent state space model. Finally, some numerical examples will be given. [ABSTRACT FROM AUTHOR]

This paper presents a methodology for the identification of high spatial density mode shapes of a dynamical system under harmonic excitation that use information from a mobile sensor. The methodology is verified numerically and experimentally with uniform simply supported beams under sinusoidal excitation. High correlation is found between the theoretical and the numerically identified mode shapes in the simulations. High correlation is also obtained between the operational mode shape identified using a traditional modal identification approach and the one using the proposed methodology during the experimental validation. The methodology minimizes errors in the identification process due to the influence of the non-resonant modes and it is robust to uncertainties in the initial identification of the natural frequencies. [ABSTRACT FROM AUTHOR]

Models used in simulation may give accurate shortterm trajectories but distort long-term (statistical) properties. In this work, we augment a given approximate model with a control law (a 'thermostat') that gently perturbs the dynamical system to target a thermodynamic state consistent with a set of prescribed (possibly evolving) observations. As proof of concept, we provide an example involving a point vortex fluid model on the sphere, for which we show convergence of equilibrium quantities (in the stationary case) and the ability of the thermostat to dynamically track a transient state. [ABSTRACT FROM AUTHOR]

The central theme of complex systems research is to understand the emergent macroscopic properties of a system from the interplay of its microscopic constituents. The emergence of macroscopic properties is often intimately related to the structure of the microscopic interactions. Here, we present an analytical approach for deriving necessary conditions that an interaction network has to obey in order to support a given type of macroscopic behaviour. The approach is based on a graphical notation, which allows rewriting Jacobi's signature criterion in an interpretable form and which can be applied to many systems of symmetrically coupled units. The derived conditions pertain to structures on all scales, ranging from individual nodes to the interaction network as a whole. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. The results complete and extend the previous analysis of Do et al. (2012Phys. Rev. Lett.108, 194102). [ABSTRACT FROM AUTHOR]

We consider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. The Central Limit Theorem for random dynamical systems is established. [ABSTRACT FROM AUTHOR]

In this paper, we develop two classes of robust preconditioners for the structure-preserving discretization of the incompressible magnetohydrodynamics (MHD) system. By studying the well-posedness of the discrete system, we design block preconditioners for them and carry out rigorous analysis on their performance. We prove that such preconditioners are robust with respect to most physical and discretization parameters. In our proof, we improve the existing estimates of the block triangular preconditioners for saddle point problems by removing the scaling parameters, which are usually difficult to choose in practice. This new technique is applicable not only to the MHD system, but also to other problems. Moreover, we prove that Krylov iterative methods with our preconditioners preserve the divergence-free condition exactly, which complements the structure-preserving discretization. Another feature is that we can directly generalize this technique to other discretizations of the MHD system. We also present preliminary numerical results to support the theoretical results and demonstrate the robustness of the proposed preconditioners. [ABSTRACT FROM AUTHOR]

We investigate the multi-valued responses of a non-linear vibro-impact oscillator with a one-sided barrier subject to random narrow-band excitation. The frequency response of the system is obtained using the Krylov-Bogoliubov averaging method. Meanwhile, the backbone curve and the critical equation of unstable region are also derived for the deterministic case. Then, the method of moment is applied to obtain the iterative calculation equation for the mean-square response amplitude under the stochastic case. Excitation frequency, nonlinearity intensity, damping parameters, especially the distance between the system’s static equilibrium position and the barrier can lead to triple-valued response under certain case. In some conditions the impact system may have two or four steady-state solutions, which is an interesting phenomenon for impact system. The unstable region is one uniform part while under smaller nonlinearity intensity it is divided into two parts. Moreover, we also find that as random noise intensity increases, the pervasion of the phase trajectories strengthens, and then destroys the topological property of the phase trajectories. [ABSTRACT FROM AUTHOR]

Here, we introduce a statistical approach derived from dynamics, for the study of the geophysical fluid dynamics phenomena characterized by a weak interaction among the variables of interest and the rest of the system. The approach is reminiscent of the one developed some years ago [M. Bianucci, R. Mannella, P. Grigolini and B. J. West, Phys. Rev. E 51, 3002 (1995)] to derive statistical mechanics of macroscopic variables on interest starting from Hamiltonian microscopic dynamics. However, in the present work, we are interested to generalize this approach beyond the context of the foundation of thermodynamics, in fact, we take into account the cases where the system of interest could be non-Hamiltonian (dissipative) and also the interaction with the irrelevant part can be of a more general type than Hamiltonian. As such example, we will refer to a typical case from geophysical fluid dynamics: the complex ocean-atmosphere interaction that gives rise to the El Niño Southern Oscillation (ENSO). Here, changing all the scales, the role of the 'microscopic' system is played by the atmosphere, while the ocean (or some ocean variables) plays the role of the intrinsically dissipative macroscopic system of interest. Thus, the chaotic and divergent features of the fast atmosphere dynamics remains in the decaying properties of the correlation functions and of the response function of the atmosphere variables, while the exponential separation of the perturbed (or close) single trajectories does not play a direct role. In the present paper, we face this problem in the frame of a not formal Langevin approach, limiting our discussion to physically based rather than mathematics arguments. Elsewhere, we obtain these results via a much more formal procedure, using the Zwanzing projection method and some elements from the Lie Algebra field. [ABSTRACT FROM AUTHOR]

The Krieger generator theorem says that every invertible ergodic measure-preserving system with finite measure-theoretic entropy can be embedded into a full shift with strictly greater topological entropy. We extend Krieger's theorem to include toral automorphisms and, more generally, any topological dynamical system on a compact metric space that satisfies almost weak specification, asymptotic entropy expansiveness, and the small boundary property. As a corollary, one obtains a complete solution to a natural generalization of an open problem in Halmos's 1956 book regarding an isomorphism invariant that he proposed. [ABSTRACT FROM AUTHOR]

We begin by considering second order dynamical systems of the from ẍ(t)+γ(t)ẋ(t)+λ(t)B(x(t))= 0, where B: H → H is a cocoercive operator defined on a real Hilbert space H, λ:[0,+∞)→[0,+∞) is a relaxation function and γ:[0,+∞)→[0,+∞) a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator B. The framework allows to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when B is the gradient of a smooth convex function the value of the latter converges along the ergodic trajectory to its minimal value with a rate of O(1/t). [ABSTRACT FROM AUTHOR]

We study the structure of attractors in the two-dimensional dynamical systemthat appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parametersµandΘ. [ABSTRACT FROM PUBLISHER]

In this paper, we prove the class of dynamical system is a variety. Also the variety of dynamical systems can be seen as a category. Then in this area we pay attention to some limits, colimits and their existence or lack of existence in the mentioned category. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of properties of-limit sets of non-autonomous dynamical systems on compact metric spaces given by sequences of maps which uniformly converge to a continuous mapf. We show that, for systems defined on compact metric spaces, if an-limit setof the non-autonomous system is a subset of the setP(f) of periodic points offthenis necessarily the union of finitely many disjoint connected sets which are cyclically mapped to one another. Using this result, we answer a question posed by Cánovas in [3] [On-limit sets of non-autonomous systems. J. Difference Equ. Appl. 12 (2006), pp. 95–100] by proving that, if an interval mapfhas only finite-limit sets, then any-limit setof the non-autonomous system is a subset of the set of periodic points off. We also show that a similar result applies to systems on trees but not on graphs with loops. [ABSTRACT FROM PUBLISHER]

Numerical computations of ground and excited states of Bose–Einstein condensates (BECs) require high spatial resolution due to the presence of the multiscale structures and boundary layers and interior layers in the solution. In this paper, a new discontinuous Galerkin (DG) method is presented for the computation of ground and excited states of BECs. For the spatial discretization, the direct discontinuous Galerkin (DDG) method is applied by using the normalized gradient flow. For the time discretization, we integrate the ordinary differential equations (ODEs) which is obtained by linearizing the interaction term. To evaluate the matrix exponential operator efficiently, we apply the Krylov subspace approximations to the matrix exponential operator. Numerical examples with different potentials are reported to demonstrate the validity and effectiveness of the semi-implicit DG method. [ABSTRACT FROM AUTHOR]

In recent years, a great interest has been shown towards Krylov subspace techniques applied to model order reduction of large-scale dynamical systems. A special interest has been devoted to single-input single-output (SISO) systems by using moment matching techniques based on Arnoldi or Lanczos algorithms. In this paper, we consider multiple-input multiple-output (MIMO) dynamical systems and introduce the rational block Arnoldi process to design low order dynamical systems that are close in some sense to the original MIMO dynamical system. Rational Krylov subspace methods are based on the choice of suitable shifts that are selected a priori or adaptively. In this paper, we propose an adaptive selection of those shifts and show the efficiency of this approach in our numerical tests. We also give some new block Arnoldi-like relations that are used to propose an upper bound for the norm of the error on the transfer function. [ABSTRACT FROM AUTHOR]

Omega-limit sets play an important role in one-dimensional dynamics. During last fifty year at least three definitions of basic set has appeared. Authors often use results with different definition. Here we fill in the gap of missing proof of equivalency of these definitions. Using results on basic sets we generalize results in paper [P. Oprocha, Invariant scrambled sets and distributional chaos, Dyn. Syst. 24 (2009), no. 1, 31–43.] to the case continuous maps of finite graphs. The Li-Yorke chaos is weaker than positive topological entropy. The equivalency arises when we add condition of invariance to Li-Yorke scrambled set. In this note we show that for a continuous graph map properties positive topological entropy; horseshoe; invariant Li-Yorke scrambled set; uniform invariant distributional chaotic scrambled set and distributionaly chaotic pair are mutually equivalent. [ABSTRACT FROM PUBLISHER]

The recent method of integration of non-stationary dynamical systems admitting nonlinear superpositions is applied to the three-dimensional dynamical systems associated with three-dimensional Vessiot-Guldberg-Lie algebras L 3 . The investigation is based on Bianchi’s classification of real three-dimensional Lie algebras and realizations of these algebras in the three-dimensional space. Enumeration of the Vessiot-Guldberg-Lie algebras L 3 allows to classify three-dimensional dynamical systems admitting nonlinear superpositions into thirty one standard types by introducing canonical variables. Twenty four of them are associated with solvable Vessiot-Guldberg-Lie algebras and can be reduced to systems of first-order linear equations. The remaining seven standard types are nonlinear. Integration of the latter types is an open problem. [ABSTRACT FROM AUTHOR]

We employ Anderson extrapolation to accelerate the classical Jacobi iterative method for large, sparse linear systems. Specifically, we utilize extrapolation at periodic intervals within the Jacobi iteration to develop the Alternating Anderson–Jacobi (AAJ) method. We verify the accuracy and efficacy of AAJ in a range of test cases, including nonsymmetric systems of equations. We demonstrate that AAJ possesses a favorable scaling with system size that is accompanied by a small prefactor, even in the absence of a preconditioner. In particular, we show that AAJ is able to accelerate the classical Jacobi iteration by over four orders of magnitude, with speed-ups that increase as the system gets larger. Moreover, we find that AAJ significantly outperforms the Generalized Minimal Residual (GMRES) method in the range of problems considered here, with the relative performance again improving with size of the system. Overall, the proposed method represents a simple yet efficient technique that is particularly attractive for large-scale parallel solutions of linear systems of equations. [ABSTRACT FROM AUTHOR]

Every orbit of a rigid rotation of a circle by a fixed irrational angle is dense. However, the apparent uniformity of the distribution of iterates after a finite number of iterations appears strikingly different for various choices of a rotation angle. Motivated by this observation, we introduce a scalar function on the orbits of a discrete dynamical system defined on a bounded metric space, called the linear limit density, which we interpret as a measure of an orbit's approach to density. Utilizing the three-distance theorem, we compute the exact value of the linear limit density of orbits of rigid rotations by irrational rotation angles with period-1 continued fraction expansions. We further show that any discrete dynamical system defined by an orientation-preserving diffeomorphism of the circle has an orbit with a larger linear limit density than any orbit of the rigid rotation by the golden number. Bernoulli shift maps acting on sequences over a finite alphabet provide another illustrative class of dynamical systems with dense orbits. Our study of the efficiency of an orbit's approach to density leads us to demonstrate the existence of a class of infinite sequences with finite linear limit density constructed by recursively extending finite de Bruijn sequences. [ABSTRACT FROM AUTHOR]

Most physical systems in reality exhibit a nonlinear relationship between input and output variables. This nonlinearity can manifest itself in terms of piecewise continuous functions or bifurcations, between some or all of the variables. The aims of this paper are two-fold. Firstly, a mixture of experts (MoE) model was trained on different physical systems exhibiting these types of nonlinearities. MoE models separate the input space into homogeneous regions and a different expert is responsible for the different regions. In this paper, the experts were low order polynomial regression models, thus avoiding the need for high-order polynomials. The model was trained within a Bayesian framework using variational Bayes, whereby a novel approach within the MoE literature was used in order to determine the number of experts in the model. Secondly, Bayesian sensitivity analysis (SA) of the systems under investigation was performed using the identified probabilistic MoE model in order to assess how uncertainty in the output can be attributed to uncertainty in the different inputs. The proposed methodology was first tested on a bifurcating Duffing oscillator, and it was then applied to real data sets obtained from the Tamar and Z24 bridges. In all cases, the MoE model was successful in identifying bifurcations and different physical regimes in the data by accurately dividing the input space; including identifying boundaries that were not parallel to coordinate axes. [ABSTRACT FROM AUTHOR]

We propose a new method for constructing partially hyperbolic diffeomorphisms on closed manifolds. As a demonstration of the method we show that there are simply connected closed manifolds that support partially hyperbolic diffeomorphisms. Laying aside many surgery constructions of 3-dimensional Anosov flows, these are the first new examples of manifolds which admit partially hyperbolic diffeomorphisms in the past forty years. [ABSTRACT FROM AUTHOR]

We look at solving large nonsymmetric systems of linear equations using polynomial preconditioned Krylov methods. We give a simple way to find the polynomial. It is shown that polynomial preconditioning can significantly improve restarted GMRES for difficult problems, and the reasons for this are examined. Stability is discussed, and algorithms are given for increased stability. Next, we apply polynomial preconditioning to GMRES with deflated restarting. It is shown that this is worthwhile for sparse matrices and for problems with many small eigenvalues. Multiple right-hand sides are also considered. [ABSTRACT FROM AUTHOR]

Identifying and characterizing geometric structure in the flow of a nonlinear dynamical system can facilitate understanding, model simplification, and solution approximation. The approach addressed in this paper uses information from finite-time Lyapunov exponents and vectors associated with the tangent linear dynamics. We refer to this approach as finite-time Lyapunov analysis (FTLA). FTLA identifies the potential for flow structure based on the stability and timescales implied by the spectrum of finite-time Lyapunov exponents. The corresponding Lyapunov vectors provide a basis for representing a splitting of the tangent space at phase points consistent with the splitting of the spectrum. A key property that makes FTLA viable is the exponential convergence of the splitting as the finite time increases. Tangency conditions for the vector field are used to determine points on manifolds of interest. The benefits of the FTLA approach are the dynamical model need not be in a special normal form, the manifolds of interest need not be attracting nor of known dimension, and the manifolds need not be associated with a fixed point or periodic orbit. After a brief review of FTLA, it is applied to spacecraft stationkeeping around a libration point in the circular restricted three-body problem. This application requires locating the stable and unstable subspaces at points on periodic and aperiodic orbits. For the periodic case, the FTLA subspaces are shown to agree with the Floquet subspaces; for the quasi-periodic case, the accuracy of the FTLA subspaces is demonstrated by simulation. [ABSTRACT FROM AUTHOR]

In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called a parallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes. [ABSTRACT FROM AUTHOR]

Complex graphics of dynamical system have striking features of fractals and become a wide area of research due to their beauty and complexity of their nature. The aim of this paper is to study dynamics of relative superior tricorns and multicorns using S-iteration schemes. Several examples are presented to explore the geometry of relative superior tricorns and multicorns for antipolynomial z→z¯n+c of complex polynomial zn+c for n≥2. [ABSTRACT FROM AUTHOR]

Summary: Modeling of dynamical systems using ordinary differential equations is a popular approach in the field of systems biology. Two of the most critical steps in this approach are to construct dynamical models of biochemical reaction networks for large datasets and complex experimental conditions and to perform efficient and reliable parameter estimation for model fitting. We present a modeling environment for MATLAB that pioneers these challenges. The numerically expensive parts of the calculations such as the solving of the differential equations and of the associated sensitivity system are parallelized and automatically compiled into efficient C code. A variety of parameter estimation algorithms as well as frequentist and Bayesian methods for uncertainty analysis have been implemented and used on a range of applications that lead to publications. Availability and implementation: The Data2Dynamics modeling environment is MATLAB based, open source and freely available at http://www.data2dynamics.org. [ABSTRACT FROM AUTHOR]

The spontaneous formation of patterns in dynamical systems is a rich phenomenon that transcends scientific boundaries. Here, we report our observation of coupled optical–atomic pattern formation, which results in the creation of self-organized, multimode structures in free-space laser-driven cold atoms. We show that this process gives rise to spontaneous three-dimensional Sisyphus cooling even at very low light intensities and the emergence of self-organized structures on both sub- and super-wavelength scales. [ABSTRACT FROM AUTHOR]

Typically, in the dynamical theory of extremal events, the function that gauges the intensity of a phenomenon is assumed to be convex and maximal, or singular, at a single, or at most a finite collection of points in phase–space. In this paper we generalize this situation to fractal landscapes, i.e. intensity functions characterized by an uncountable set of singularities, located on a Cantor set. This reveals the dynamical rôle of classical quantities like the Minkowski dimension and content, whose definition we extend to account for singular continuous invariant measures. We also introduce the concept of extremely rare event, quantified by non-standard Minkowski constants and we study its consequences to extreme value statistics. Limit laws are derived from formal calculations and are verified by numerical experiments. [ABSTRACT FROM AUTHOR]