Dynamical Systems Point Of View Research Articles

Center manifolds for a class of degenerate evolution equations and existence of small-amplitude kinetic shocks

We construct center manifolds for a class of degenerate evolution equations including the steady Boltzmann equation and related kinetic models, establishing in the process existence and behavior of small-amplitude kinetic shock and boundary layers. Notably, for Boltzmann's equation, we show that elements of the center manifold decay in velocity at near-Maxwellian rate, in accord with the formal Chapman–Enskog picture of near-equilibrium flow as evolution along the manifold of Maxwellian states, or Grad moment approximation via Hermite polynomials in velocity. Our analysis is from a classical dynamical systems point of view, with a number of interesting modifications to accommodate ill-posedness of the underlying evolution equation.

Physical Reservoir Computingの数理

Physical Reservoir Computingの数理

A new description of epileptic seizures based on dynamic analysis of a thalamocortical model

Increasing evidence suggests that the brain dynamics can be interpreted from the viewpoint of nonlinear dynamical systems. The aim of this paper is to investigate the behavior of a thalamocortical model from this perspective. The model includes both cortical and sensory inputs that can affect the dynamic nature of the model. Driving response of the model subjected to various harmonic stimulations is considered to identify the effects of stimulus parameters on the cortical output. Detailed numerical studies including phase portraits, Poincare maps and bifurcation diagrams reveal a wide range of complex dynamics including period doubling and chaos in the output. Transition between different states can occur as the stimulation parameters are changed. In addition, the amplitude jump phenomena and hysteresis are shown to be possible as a result of the bending in the frequency response curve. These results suggest that the jump phenomenon due to the brain nonlinear resonance can be responsible for the transitions between ictal and interictal states.

A dynamical systems framework for crop models: Toward optimal fertilization and irrigation strategies under climatic variability

Crop models are widely used for the modeling and prediction of crop yields, as decision support tools, and to develop research questions. Though typically constructed as a set of dynamical equations, crop models are not often analyzed from a specifically dynamical systems point of view, despite its potential to elucidate the roles of feedbacks and internal and external forcings on system stability and the optimization of control protocols (e.g., irrigation and fertilization). Here we develop a minimal dynamical system, based in part on the widely known AquaCrop model, consisting of a set of ordinary differential equations (ODE's) describing the evolution of canopy cover, soil moisture, and soil nitrogen. These state variables are coupled through canopy growth and senescence, the evapotranspiration and percolation of soil moisture, and the uptake and leaching of soil nitrogen. The system is driven by random hydroclimatic forcing. Important crop model responses, such as biomass and yield, are calculated, and optimal yield and profitability under differing climate scenarios, irrigation strategies, and fertilization strategies are examined within the developed framework. The results highlight the need to maintain the system at or above resource limitation thresholds to achieve optimality and the role of system variability in determining management strategies.

Dynamics of relations associated with two core decompositions

Given an unshielded continuum K⊂C, the core decomposition DKLC of K with respect to local connectedness is known to exist [2]. Such a decomposition DKLC is monotone, locally connected under quotient topology, and refines every other monotone decomposition D′ which is locally connected under quotient topology. Let ∼ be the closed equivalence whose classes form the decomposition DKLC. Then ∼ contains a symmetric closed relation RK which requires (x, y) ∈ RK if and only if x and y lie in a prime end impression of C∖K. We also say that the relation RK respects prime end impressions. From Akin’s viewpoint of dynamical systems, the equivalence ∼ may be obtained as one of the limit relations of RK, through transfinite process. We will propose a direct approach to realize this limit relation in a concrete construction. More or less, such an approach builds the elements of DKLC “from below ” . If K is an unshielded disconnected compactum, the core decomposition DKFS of K with respect to the finitely suslinian property has been obtained in [3]. In this case, we consider a symmetric closed relation that “respects limit continua” and show that the same approach also works, in constructing DKFS from below.

Complex Dynamic Systems View on Conceptual Change: How a Picture of Students’ Intuitive Conceptions Accrue From Dynamically Robust Task Dependent Learning Outcomes

We discuss here conceptual change and the formation of robust learning outcomes from the viewpoint of complex dynamic systems (CDS). The CDS view considers students’ conceptions as context dependent and multifaceted structures which depend on the context of their application. In the CDS view the conceptual patterns (i.e. intuitive conceptions here) may be robust in a certain situation but are not formed, at least not as robust ones, in another situation. The stability is then thought to arise dynamically in a variety of ways and not so much to mirror rigid ontological categories or static intuitive conceptions. We use computational modelling to understand the generic dynamic and emergent features of that phenomenon. The model is highly simplified and idealized, but it shows how context dependence, described here by an epistemic landscape structure, leads to the formation of context dependent robust states that can be viewed as attractors in learning, and how owing to the sharply defined nature of these states, learning appears as a progression of switches from one state to another, giving thus the appearance of conceptual change as switches from one robust state to another. Finally, we discuss the implications of the results in directing attention to the design of learning tasks and their structure, and how empirically accessible learning outcomes might be related to these underlying factors.

A random dynamical systems perspective on stochastic resonance

We study stochastic resonance in an over-damped approximation of the stochastic Duffing oscillator from a random dynamical systems point of view. We analyse this problem in the general framework of random dynamical systems with a nonautonomous forcing. We prove the existence of a unique global attracting random periodic orbit and a stationary periodic measure. We use the stationary periodic measure to define an indicator for the stochastic resonance.

Order and chaos in the one-dimensional ϕ4 model: N-dependence and the Second Law of Thermodynamics

We revisit the equilibrium one-dimensional ϕ4 model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These “computationally stable” sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles’ coordinates and momenta are either identical or differ only in sign. “Positive Lyapunov exponents” can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results.We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model’s chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.

Dynamics and spectral theory of quasi-periodic Schrödinger-type operators

We survey the theory of quasi-periodic Schrödinger-type operators, focusing on the advances made since the early 2000s by adopting a dynamical systems point of view.

The Painlevé paradox in contact mechanics

The 120-year old so-called Painleve paradox involves the loss of determinism in models of planar rigid bodies in point contact with a rigid surface, subject to Coulomb-like dry friction. The phenomenon occurs due to coupling between normal and rotational degrees-of-freedom such that the effective normal force becomes attractive rather than repulsive. Despite a rich literature, the forward evolution problem remains unsolved other than in certain restricted cases in 2D with single contact points. Various practical consequences of the theory are revisited, including models for robotic manipulators, and the strange behaviour of chalk when pushed rather than dragged across a blackboard. Reviewing recent theory, a general formulation is proposed, including a Poisson or energetic impact law. The general problem in 2D with a single point of contact is discussed and cases or inconsistency or indeterminacy enumerated. Strategies to resolve the paradox via contact regularisation are discussed from a dynamical systems point of view. By passing to the infinite stiffness limit and allowing impact without collision, inconsistent and indeterminate cases are shown to be resolvable for all open sets of conditions. However, two unavoidable ambiguities that can be reached in finite time are discussed in detail, so called dynamic jam and reverse chatter. A partial review is given of 2D cases with two points of contact showing how a greater complexity of inconsistency and indeterminacy can arise. Extension to fully three-dimensional analysis is briefly considered and shown to lead to further possible singularities. In conclusion, the ubiquity of the Painleve paradox is highlighted and open problems are discussed.

Constructing networks from a dynamical system perspective for multivariate nonlinear time series.

We describe a method for constructing networks for multivariate nonlinear time series. We approach the interaction between the various scalar time series from a deterministic dynamical system perspective and provide a generic and algorithmic test for whether the interaction between two measured time series is statistically significant. The method can be applied even when the data exhibit no obvious qualitative similarity: a situation in which the naive method utilizing the cross correlation function directly cannot correctly identify connectivity. To establish the connectivity between nodes we apply the previously proposed small-shuffle surrogate (SSS) method, which can investigate whether there are correlation structures in short-term variabilities (irregular fluctuations) between two data sets from the viewpoint of deterministic dynamical systems. The procedure to construct networks based on this idea is composed of three steps: (i) each time series is considered as a basic node of a network, (ii) the SSS method is applied to verify the connectivity between each pair of time series taken from the whole multivariate time series, and (iii) the pair of nodes is connected with an undirected edge when the null hypothesis cannot be rejected. The network constructed by the proposed method indicates the intrinsic (essential) connectivity of the elements included in the system or the underlying (assumed) system. The method is demonstrated for numerical data sets generated by known systems and applied to several experimental time series.

Numerical approximation of the non-essential spectrum of abstract delay differential equations

Abstract delay differential equations (ADDEs) extend delay differential equations (DDEs) from finite to infinite dimension. They arise in many application fields. From a dynamical system point of view, the stability analysis of an equilibrium is the first relevant question, which can be reduced to the stability of the zero solution of the corresponding linearized system. In the understanding of the linear case, the essential and the non-essential spectra of the infinitesimal generator are crucial. We propose to extend the infinitesimal generator approach developed for linear DDEs to approximate the non-essential spectrum of linear ADDEs. We complete the paper with the numerical results for a homogeneous neural field model with transmission delay of a single population of neurons.

Multistable jittering in oscillators with pulsatile delayed feedback.

Oscillatory systems with time-delayed pulsatile feedback appear in various applied and theoretical research areas, and received a growing interest in recent years. For such systems, we report a remarkable scenario of destabilization of a periodic regular spiking regime. At the bifurcation point numerous regimes with nonequal interspike intervals emerge. We show that the number of the emerging, so-called "jittering" regimes grows exponentially with the delay value. Although this appears as highly degenerate from a dynamical systems viewpoint, the "multijitter" bifurcation occurs robustly in a large class of systems. We observe it not only in a paradigmatic phase-reduced model, but also in a simulated Hodgkin-Huxley neuron model and in an experiment with an electronic circuit.

Renormalization Method in p-Adic λ-Model on the Cayley Tree

In this present paper, it is proposed the renormalization techniques in the investigation of phase transition phenomena in $p$-adic statistical mechanics. We mainly study $p$-adic $\l$-model on the Cayley tree of order two. We consider generalized $p$-adic quasi Gibbs measures depending on parameter $\r\in\bq_p$, for the $\l$-model. Such measures are constructed by means of certain recurrence equations. These equations define a dynamical system. We study two regimes with respect to parameters. In the first regime we establish that the dynamical system has one attractive and two repelling fixed points, which predicts the existence of a phase transition. In the second regime the system has two attractive and one neutral fixed points, which predicts the existence of a quasi phase transition. A main point of this paper is to verify (i.e. rigorously prove) and confirm that the indicated predictions (via dynamical systems point of view) are indeed true. To establish the main result, we employ the methods of $p$-adic analysis, and therefore, our results are not valid in the real setting.

Visualization of Input Parameters for Stream and Pathline Seeding

Uncertainty arises in all stages of the visualization pipeline. However, the majority of flow visualization applications convey no uncertainty information to the user. In tools where uncertainty is conveyed, the focus is generally on data, such as error that stems from numerical methods used to generate a simulation or on uncertainty associated with mapping visualiza-tion primitives to data. Our work is aimed at another source of uncertainty - that associated with user-controlled input param-eters. The navigation and stability analysis of user-parameters has received increasing attention recently. This work presents an investigation of this topic for flow visualization, specifically for three-dimensional streamline and pathline seeding. From a dynamical systems point of view, seeding can be formulated as a predictability problem based on an initial condition. Small perturbations in the initial value may result in large changes in the streamline in regions of high unpredictability. Analyzing this predictability quantifies the perturbation a trajectory is subjugated to by the flow. In other words, some predictions are less certain than others as a function of initial conditions. We introduce novel techniques to visualize important user input parameters such as streamline and pathline seeding position in both space and time, seeding rake position and orientation, and inter-seed spacing. The implementation is based on a metric which quantifies similarity between stream and pathlines. This is important for Computational Fluid Dynamics (CFD) engineers as, even with the variety of seeding strategies available, manual seeding using a rake is ubiquitous. We present methods to quantify and visualize the effects that changes in user-controlled input parameters have on the resulting stream and pathlines. We also present various visualizations to help CFD scientists to intuitively and effectively navigate this parameter space. The reaction from a domain expert in fluid dynamics is also reported.

An analysis on the asymptotic behavior of multistep linearly implicit schemes for the Duffing equation

We consider the discrete gradient method for dissipative lineargradient systems, which strictly replicates the dissipation property, yielding a remarkable stability. However, it also replicates the nonlinearity of an original equation. To overcome this, we can employ multistep linearly implicit schemes as a relaxation; however, it can in turn destroy the originally aimed stability. Matsuo–Furihata (2014) introduced a dynamical systems viewpoint to understand the behavior for a toy scalar problem. In this letter, we show that their method can work also for the two-dimensional Duffing equation. There a new concept of semi-strong Lyapunov functionals is required.

Variable threshold algorithm for division of labor analyzed as a dynamical system.

Division of labor is a widely studied aspect of colony behavior of social insects. Division of labor models indicate how individuals distribute themselves in order to perform different tasks simultaneously. However, models that study division of labor from a dynamical system point of view cannot be found in the literature. In this paper, we define a division of labor model as a discrete-time dynamical system, in order to study the equilibrium points and their properties related to convergence and stability. By making use of this analytical model, an adaptive algorithm based on division of labor can be designed to satisfy dynamic criteria. In this way, we have designed and tested an algorithm that varies the response thresholds in order to modify the dynamic behavior of the system. This behavior modification allows the system to adapt to specific environmental and collective situations, making the algorithm a good candidate for distributed control applications. The variable threshold algorithm is based on specialization mechanisms. It is able to achieve an asymptotically stable behavior of the system in different environments and independently of the number of individuals. The algorithm has been successfully tested under several initial conditions and number of individuals.

Escape dynamics of many hard disks.

Many-particle effects in escapes of hard disks from a square box via a hole are discussed in a viewpoint of dynamical systems. Starting from N disks in the box at the initial time, we calculate the probability P_{n}(t) for at least n disks to remain inside the box at time t for n=1,2,...,N. At early times, the probabilities P_{n}(t),n=2,3,...,N-1, are described by superpositions of exponential decay functions. On the other hand, after a long time the probability P_{n}(t) shows a power-law decay ∼t^{-2n} for n≠1, in contrast to the fact that it decays with a different power law ∼t^{-n} for cases without any disk-disk collision. Chaotic or nonchaotic properties of the escape systems are discussed by the dynamics of a finite-time largest Lyapunov exponent, whose decay properties are related with those of the probability P_{n}(t).

Dark D-brane cosmology

Disformally coupled cosmologies arise from Dirac-Born-Infeld actions in Type II string theories, when matter resides on a moving hidden sector D-brane. Since such matter interacts only very weakly with the standard model particles, this scenario can provide a natural origin for the dark sector of the universe with a clear geometrical interpretation: dark energy is identified with the scalar field associated to the D-brane's position as it moves in the internal space, acting as quintessence, while dark matter is identified with the matter living on the D-brane,which can be modelled by a perfect fluid. The coupling functions are determined by the (warped) extra-dimensional geometry, and are thus constrained by the theory. The resulting cosmologies are studied using both dynamical system analysis and numerics. From the dynamical system point of view, one free parameter controls the cosmological dynamics, given by the ratio of the warp factor and the potential energy scales. The disformal coupling allows for new scaling solutions that can describe accelerating cosmologies alleviating the coincidence problem of dark energy. In addition, this scenario may ameliorate the fine-tuning problem of dark energy, whose small value may be attained dynamically, without requiring the mass of the dark energy field to be unnaturally low.

Damping filter method for obtaining spatially localized solutions.

Spatially localized structures are key components of turbulence and other spatiotemporally chaotic systems. From a dynamical systems viewpoint, it is desirable to obtain corresponding exact solutions, though their existence is not guaranteed. A damping filter method is introduced to obtain variously localized solutions and adapted in two typical cases. This method introduces a spatially selective damping effect to make a good guess at the exact solution, and we can obtain an exact solution through a continuation with the damping amplitude. The first target is a steady solution to the Swift-Hohenberg equation, which is a representative of bistable systems in which localized solutions coexist and a model for spanwise-localized cases. Not only solutions belonging to the well-known snaking branches but also those belonging to isolated branches known as "isolas" are found with continuation paths between them in phase space extended with the damping amplitude. This indicates that this spatially selective excitation mechanism has an advantage in searching spatially localized solutions. The second target is a spatially localized traveling-wave solution to the Kuramoto-Sivashinsky equation, which is a model for streamwise-localized cases. Since the spatially selective damping effect breaks Galilean and translational invariances, the propagation velocity cannot be determined uniquely while the damping is active, and a singularity arises when these invariances are recovered. We demonstrate that this singularity can be avoided by imposing a simple condition, and a localized traveling-wave solution is obtained with a specific propagation speed.