Dynamical Systems Point Research Articles

The parameterization method for center manifolds

In this paper, we present a generalization of the parameterization method, introduced by Cabré, Fontich and De la Llave, to center manifolds associated to non-hyperbolic fixed points of discrete dynamical systems. As a byproduct, we find a new proof for the existence and regularity of center manifolds. However, in contrast to the classical center manifold theorem, our parameterization method will simultaneously obtain the center manifold and its conjugate center dynamical system. Furthermore, we will provide bounds on the error between approximations of the center manifold and the actual center manifold, as well as bounds for the error in the conjugate dynamical system.

Hybrid Automata: An Algorithmic Approach Behavioral Hybrid Systems

Hybrid automata strategies have advanced as a vital tool to design, check and direct the execution of hybrid systems. Any way they can – and we assume should – be utilized to communicate quantitative models about hybrid systems in different areas, for example, experimental sciences. Since the conventional design of hybrid automata compares well to consecutively integrate behavioral chains in living creatures, we look for a use of hybrid modeling procedures in the social sciences and, particularly, brain research. We attempt to address the question related to how human drivers move onto an expressway and simultaneously utilize this study as our test-bed for utilizing hybrid automata inside behavioral sciences. Hybrid automata give a language to displaying and exploring advanced and simple calculations in real-time systems. Hybrid automata are studied here from a dynamical systems point of view. Essential and adequate conditions for the presence and uniqueness of arrangements are inferred and a class of hybrid automata whose arrangements rely consistently upon the underlying state is described. The outcomes on presence, uniqueness, and progression fill in as a beginning stage for solid study. In this paper, we present the structure of hybrid automata as a model and detailed language for hybrid systems. Hybrid automata can be seen as a theory of timed automata, in which the behavior of factors is represented in each state by a bunch of differential conditions. We show that a large number of the models considered in the workshop can be characterized by hybrid automata. While the reachability issue is undecidable in any event, for extremely confined classes of hybrid automata, we present two semi-decision techniques for checking security properties of piecewise-straight hybrid automata, in which all factors change at steady rates. The two techniques are based, individually, on limiting and figuring fix points on for the most part endless state spaces. We show that if the end of the method, at that point they offer the right responses. We then show that for a significant number of the run of the mill workshop models, the strategies do end and hence give an algorithmic approach to confirming their properties.

Periodic orbits of discrete and continuous dynamical systems via Poincaré-Miranda theorem

We present a systematic methodology to determine and locate analytically isolated periodic points of discrete and continuous dynamical systems with algebraic nature. We apply this method to a wide range of examples, including a one-parameter family of counterexamples to the discrete Markus-Yamabe conjecture (La Salle conjecture); the study of the low periods of a Lotka-Volterra-type map; the existence of three limit cycles for a piece-wise linear planar vector field; a new counterexample of Kouchnirenko's conjecture; and an alternative proof of the existence of a class of symmetric central configuration of the $(1+4)$-body problem.

Convergence for Slow Discrete Dynamical Systems with Identity Linearization

In this work we give sufficient and necessary conditions for convergence for nonhyperbolic fixed points of dynamical systems of arbitrary dimension whose linearization around zero is the identity function. To achieve this goal, we first rewrite the dynamical system in terms of spherical polar coordinates and by approximation of the radial iteration function we discover a necessary condition depending on a remarkable angular function. Searching for conditions that are sufficient, we discover more angular functions that together with the first one gives a complete set that plays the role of the iteration derivative for unidimensional discrete systems.

Numerical Nonlinear Analysis for Dynamic Stability of an Ankle-Hip Model of Balance on a Balance Board

Abstract Study of human upright posture (UP) stability is of great relevance to fall prevention and rehabilitation, especially for those with balance deficits for whom a balance board (BB) is a widely used mechanism to improve balance. The stability of the human-BB system has been widely investigated from a dynamical system point of view. However, most studies assume small disturbances, which allow to linearize the nonlinear human-BB dynamical system, neglecting the effect of the nonlinear terms on the stability. Such assumption has been useful to simplify the system and use bifurcation analyses to determine local dynamic stability properties. However, dynamic stability analysis results through such linearization of the system have not been verified. Moreover, bifurcation analyses cannot provide insight on dynamical behaviors for different points within the stable and unstable regions. In this study, we numerically solve the nonlinear delay differential equation that describes the human-BB dynamics for a range of selected parameters (proprioceptive feedback and time-delays). The resulting solutions in time domain are used to verify the stability properties given by the bifurcation analyses and to compare different dynamical behaviors within the regions. Results show that the selected bifurcation parameters have significant impacts not only on UP stability but also on the amplitude, frequency, and increasing or decaying rate of the resulting trajectory solutions.

Special Issue: Nonlinear and Computational Dynamics in Biomedical Applications

Special Issue: Nonlinear and Computational Dynamics in Biomedical Applications

Remarks on definitions of periodic points for nonautonomous dynamical system

ABSTRACT Let be a nonautonomous dynamical system. In this paper, we summarize known definitions of periodic points for general nonautonomous dynamical systems and propose a new definition of asymptotic periodicity. This definition is not only very natural but also resistant to changes of the beginning of the sequence generating the nonautonomous system. We show the relations among these definitions and discuss their properties. We prove that for pointwise convergent nonautonomous systems topological transitivity together with a dense set of asymptotically periodic points imply sensitivity. We also show that even for uniformly convergent systems, the nonautonomous analogue of Sharkovsky's theorem is not valid for most definitions of periodic points.

Detour induced by the piston effect in the oscillatory double-diffusive convection of a near-critical fluid

The nonlinear oscillatory double-diffusive convection in a thermodynamically near-critical binary fluid layer is investigated to explore the interactions between the piston effect and natural convection in the presence of subcritical bifurcation. The bifurcation diagram of the system is studied. Two subcritical bifurcation branches are depicted, which, together with the trivial branch of pure diffusion, are connected by two hysteresis loops. To understand the role of the piston effect, the Boussinesq counterpart of the near-critical system is considered and compared. Results show that the onset of convection is significantly altered by the piston effect. For the Boussinesq system, the lower boundary layer becomes unstable, brings on finite-amplitude perturbations, and leads to a statistically steady state. However, the near-critical system features a two-stage evolution. In the first stage, the lower boundary layer becomes unstable and then returns to stability. As soon as the temperature field relaxes into the second stage, a change of criterion occurs, and the fluid becomes unstable again. The residual convection motions amplify and finally result in finite-amplitude convection. By this means, the near-critical system becomes insensitive to the existence of the higher equilibrium state in hysteresis loops, and detours relative to the Boussinesq system are observed. This paper gives new insights into the piston effect and its interactions with natural convection from a dynamic system point of view. The conclusions can be extended to other situations where subcritical bifurcations exist.

On graph algebras from interval maps

We produce and study a family of representations of relative graph algebras on Hilbert spaces that arise from the orbits of points of 1-dimensional dynamical systems, where the underlying Markov interval maps f have escape sets. We identify when such representations are faithful in terms of the transitions to the escape subintervals.

Sparse Recovery and Dictionary Learning to Identify the Nonlinear Dynamical Systems: One Step Toward Finding Bifurcation Points in Real Systems

Modeling real dynamical systems is an important challenge in many areas of science. Extracting governing equations of systems from their time-series is a possible solution for such a challenge. In this paper, we use the sparse recovery and dictionary learning to extract governing equations of a system with parametric basis functions. In this algorithm, the assumption of sparsity in the functions of dynamical equations is used. The proposed algorithm is applied to different types of discrete and continuous nonlinear dynamical systems to show the generalization ability of this method. On the other hand, transition from one dynamical regime to another is an important concept in studying real world complex systems like biological and climate systems. Lyapunov exponent is an early warning index. It can predict bifurcation points in dynamical systems. Computation of Lyapunov exponent is a major challenge in its application in real systems, since it needs long time data to be accurate. In this paper, we use the predicted governing equation to generate long time-series, which is needed for Lyapunov exponent calculation. So the proposed method can help us to predict bifurcation points by accurate calculation of Lyapunov exponents.

An overview of the extremal index.

For a wide class of stationary time series, extreme value theory provides limiting distributions for rare events. The theory describes not only the size of extremes but also how often they occur. In practice, it is often observed that extremes cluster in time. Such short-range clustering is also accommodated by extreme value theory via the so-called extremal index. This review provides an introduction to the extremal index by working through a number of its intuitive interpretations. Thus, depending on the context, the extremal index may represent (i) the loss of independently and identically distributed degrees of freedom, (ii) the multiplicity of a compound Poisson point process, and (iii) the inverse mean duration of extreme clusters. More recently, the extremal index has also been used to quantify (iv) recurrences around unstable fixed points in dynamical systems.

On the overfly algorithm in deep learning of neural networks

In this paper, we investigate the supervised backpropagation training of multilayer neural networks from a dynamical systems point of view. We discuss some links with the qualitative theory of differential equations and introduce the overfly algorithm to tackle the local minima problem. Our approach is based on the existence of first integrals of the generalised gradient system with build-in dissipation.

D-Chain Tomography of Networks: A New Structure Spectrum and an Application to the SIR Process

The analysis of the dynamics on complex networks is closely connected to structural features of the networks. Features like, for instance, graph-cores and node degrees have been studied ubiquitously. Here we introduce the D-spectrum of a network, a novel new framework that is based on a collection of nested chains of subgraphs within the network. Graph-cores and node degrees are merely from two particular such chains of the D-spectrum. Each chain gives rise to a ranking of nodes and, for a fixed node, the collection of these ranks provides us with the D-spectrum of the node. Besides a node deletion algorithm, we discover a connection between the D-spectrum of a network and some fixed points of certain graph dynamical systems (MC systems) on the network. Using the D-spectrum we identify nodes of similar spreading power in the susceptible-infectious-recovered (SIR) model on a collection of real world networks as a quick application. We then discuss our results and conclude that D-spectra represent a meaningful augmentation of graph-cores and node degrees.

Tautness for sets of multiples and applications to ${\mathcal B}$-free dynamics

For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its \emph{set of multiples} $\mathcal M_{\mathcal B}:=\bigcup_{b\in\mathcal B}b\mathbb Z$ and the set of \emph{$\mathcal B$-free numbers} $\mathcal F_{\mathcal B}:=\mathbb Z\setminus\mathcal M_{\mathcal B}$. Tautness of the set $\mathcal B$ is a basic property related to questions around the asymptotic density of $\mathcal M_{\mathcal B}\subseteq\mathbb Z$. From a dynamical systems point of view (originated by Sarnak) one studies $\eta$, the indicator function of $\mathcal F_{\mathcal B}\subseteq\mathbb Z$, its shift-orbit closure $X_\eta\subseteq\{0,1\}^{\mathbb Z}$ and the stationary probability measure $\nu_\eta$ defined on $X_\eta$ by the frequencies of finite blocks in $\eta$. In this paper we prove that tautness implies the following two properties of $\eta$: (1) The measure $\nu_\eta$ has full topological support in $X_\eta$. (2) If $X_\eta$ is proximal, i.e. if the one-point set $\{\dots000\dots\}$ is contained in $X_\eta$ and is the unique minimal subset of $X_\eta$, then $X_\eta$ is hereditary, i.e. if $x\in X_\eta$ and if $w$ is an arbitrary element of $\{0,1\}^{\mathbb Z}$, then also the coordinate-wise product $w\cdot x$ belongs to $X_\eta$. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that $\mathcal B$ has light tails for the same conclusions.

Research of the stability of some hereditary dynamic systems

In the training course of the theory of differential equations, there exists a section on the investigation of the stability of systems of differential equations. If the system of differential equations consists of differential equations of integer order, then the stability theory of Lyapunov is usually used to research the stability of their rest points. However, in the case when the system of differential equations consists of differential equations of non-integer order, then it is necessary to use other methods of investigating the stability of such systems. Therefore, this article is devoted to the method of investigating the rest points of systems of differential equations of fractional order. In this paper we will investigate the stability of the rest points of the hereditary dynamical systems by the example of some fractal oscillators. Moreover, we will consider two types of hereditary dynamical systems: commensurable and incommensurate, for which the corresponding stability theorems for rest points are valid. Next, examples of applying these stability theorems to a fractal linear oscillator, the Duffing fractal oscillator, will be considered. The results of the research of the stability of the rest points of the hereditary dynamical systems were confirmed by constructing phase trajectories for the fractal oscillators under consideration. This article can be useful in the study of a fairly new section in the theory of differential equations-fractional calculus.

Reduced carbon cycle resilience across the Palaeocene–Eocene Thermal Maximum

Abstract. Several past episodes of rapid carbon cycle and climate change are hypothesised to be the result of the Earth system reaching a tipping point beyond which an abrupt transition to a new state occurs. At the Palaeocene–Eocene Thermal Maximum (PETM) at ∼56 Ma and at subsequent hyperthermal events, hypothesised tipping points involve the abrupt transfer of carbon from surface reservoirs to the atmosphere. Theory suggests that tipping points in complex dynamical systems should be preceded by critical slowing down of their dynamics, including increasing temporal autocorrelation and variability. However, reliably detecting these indicators in palaeorecords is challenging, with issues of data quality, false positives, and parameter selection potentially affecting reliability. Here we show that in a sufficiently long, high-resolution palaeorecord there is consistent evidence of destabilisation of the carbon cycle in the ∼1.5 Myr prior to the PETM, elevated carbon cycle and climate instability following both the PETM and Eocene Thermal Maximum 2 (ETM2), and different drivers of carbon cycle dynamics preceding the PETM and ETM2 events. Our results indicate a loss of “resilience” (weakened stabilising negative feedbacks and greater sensitivity to small shocks) in the carbon cycle before the PETM and in the carbon–climate system following it. This pre-PETM carbon cycle destabilisation may reflect gradual forcing by the contemporaneous North Atlantic Volcanic Province eruptions, with volcanism-driven warming potentially weakening the organic carbon burial feedback. Our results are consistent with but cannot prove the existence of a tipping point for abrupt carbon release, e.g. from methane hydrate or terrestrial organic carbon reservoirs, whereas we find no support for a tipping point in deep ocean temperature.

Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks

We construct stable manifolds for a class of singular evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium states. Our analysis is from a classical dynamical systems point of view, but with a number of interesting modifications to accomodate ill-posedness with respect to the Cauchy problem of the underlying evolution equation.

Predicting tipping points of dynamical systems during a period-doubling route to chaos.

Classical indicators of tipping points have limitations when they are applied to an ecological and a biological model. For example, they cannot correctly predict tipping points during a period-doubling route to chaos. To counter this limitation, we here try to modify four well-known indicators of tipping points, namely the autocorrelation function, the variance, the kurtosis, and the skewness. In particular, our proposed modification has two steps. First, the dynamic of the considered system is estimated using its time-series. Second, the original time-series is divided into some sub-time-series. In other words, we separate the time-series into different period-components. Then, the four different tipping point indicators are applied to the extracted sub-time-series. We test our approach on an ecological model that describes the logistic growth of populations and on an attention-deficit-disorder model. Both models show different tipping points in a period-doubling route to chaos, and our approach yields excellent results in predicting these tipping points.

A Branch Point on Differentiation Trajectory is the Bifurcating Event Revealed by Dynamical Network Biomarker Analysis of Single-Cell Data

The advance in single-cell profiling technologies and the development in computational algorithms provide the opportunity to reconstruct pseudo temporal trajectory with branch point of cellular development. On the other hand, theories such as dynamical network biomarkers (DNB) theory have been recently proposed to characterize the pre-transition state in biological systems. Few studies have validated whether the branch point identified in pseudo time is the critical point in dynamical system. In this study, the dynamical behavior of the branch point on the pseudo trajectory has been investigated. We study the pseudo temporal trajectories reconstructed by Wishbone and diffusion pseudotime analysis (DPT) algorithms, as well as the simulated trajectory. DNB theory is applied to justify the bifurcating event on the pseudo trajectories. Our results demonstrate that the branch point recovered by Wishbone and DPT algorithms is confirmed as a transition state in cell differentiation process by DNB theory. Furthermore, we show that an appropriate DNB group will amplify the comprehensive index of critical event as defined in DNB theory. Our study provides biological insights on pseudo trajectory with branch point in a dynamical view and also indicates that DNB theory may serve as a benchmark to check the validity of branch point.

Recurrence of Transitive Points in Dynamical Systems with the Specification Property

Let T: X → X be a continuous map of a compact metric space X. A point x ∈ X is called Banach recurrent point if for all neighborhood V of x, {n ∈ ℕ: Tn(x) ∈ V} has positive upper Banach density. Denote by Tr(T), W(T), QW(T) and BR(T) the sets of transitive points, weakly almost periodic points, quasi-weakly almost periodic points and Banach recurrent points of (X, T). If (X, T) has the specification property, then we show that every transitive point is Banach recurrent and ∅ ≠ W(T) ∩ Tr(T) ⫋ W * (T) ∩ Tr(T) ⫋ QW(T) ∩ Tr(T) ⫋ BR(T) ∩ Tr(T), in which W * (T) is a recurrent points set related to an open question posed by Zhou and Feng. Specifically the set Tr(T) ∩ W * (T) \ W(T) is residual in X. Moreover, we construct a point x ∈ BR \ QW in symbol dynamical system, and demonstrate that the sets W(T),QW(T) and BR(T) of a dynamical system are all Borel sets.