One-dimensional Dynamical Systems Research Articles

Predicting transient dynamics in a model of reed musical instrument with slowly time-varying control parameter.

When playing a self-sustained reed instrument (such as the clarinet), initial acoustical transients (at the beginning of a note) are known to be of crucial importance. Nevertheless, they have been mostly overlooked in the literature on musical instruments. We investigate here the dynamic behavior of a simple model of reed instrument with a time-varying blowing pressure accounting for attack transients performed by the musician. In practice, this means studying a one-dimensional non-autonomous dynamical system obtained by slowly varying in time the bifurcation parameter (the blowing pressure) of the corresponding autonomous systems, i.e., whose bifurcation parameter is constant. In this context, the study focuses on the case for which the time-varying blowing pressure crosses the bistability domain (with the coexistence of a periodic solution and an equilibrium) of the corresponding autonomous model. Considering the time-varying blowing pressure as a new (slow) state variable, the considered non-autonomous one-dimensional system becomes an autonomous two-dimensional fast-slow system. In the bistability domain, the latter has attracting manifolds associated with two stable branches of the bifurcation diagram of the system with constant parameter. In the framework of the geometric singular perturbation theory, we show that a single solution of the two-dimensional fast-slow system can be used to describe the global system behavior. Indeed, this allows us to determine, depending on the initial conditions and rate of change of the blowing pressure, which manifold is approached when the bistability domain is crossed and to predict whether a sound is produced during transient as a function of the musician's control.

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

High-order nonstandard finite difference methods preserving dynamical properties of one-dimensional dynamical systems

Dynamic tipping and cyclic folds, in a one-dimensional non-smooth dynamical system linked to climate models

We study the behaviour at tipping points close to (smoothed) non-smooth fold bifurcations in one-dimensional oscillatory forced systems with slow parameter drift. The focus is the non-smooth fold in the Stommel 2-box, and related climate models, which are piecewise-smooth continuous dynamical systems, modelling thermohaline circulation. These exhibit non-smooth fold bifurcations which arise when a saddle-point and a focus meet at a border collision bifurcation. By using techniques from the theory of non-smooth dynamical systems we are able to provide precise estimates for the general tipping behaviour close to the non-smooth fold as parameters vary. These are different from the usual tipping point estimates at a saddle–node bifurcation, with advanced tipping apparent in the non-smooth case when compared to the behaviour near smoothed approximations to this fold. We also see rapid, and non-monotone, transitions in the tipping points for oscillatory forced systems close to both non-smooth folds and saddle–node bifurcations due to the effects of both phase changes and non-smoothness. These variations can have implications for the prediction of tipping in climate systems, particularly in close proximity to a non-smooth fold.

Stochastic paleoclimatology: Modeling the EPICA ice core climate records.

We analyze and model the stochastic behavior of paleoclimate time series and assess the implications for the coupling of climate variables during the Pleistocene glacial cycles. We examine 800 kiloyears of carbon dioxide, methane, nitrous oxide, and temperature proxy data from the European Project for Ice Coring in Antarctica (EPICA) Dome-C ice core, which are characterized by 100 ky glacial cycles overlain by fluctuations across a wide range of timescales. We quantify this behavior through multifractal time-weighted detrended fluctuation analysis, which distinguishes near-red-noise and white-noise behavior below and above the 100 ky glacial cycle, respectively, in all records. This allows us to model each time series as a one-dimensional periodic nonautonomous stochastic dynamical system, and assess the stability of physical processes and the fidelity of model-simulated time series. We extend this approach to a four-variable model with intervariable coupling terms, which we interpret in terms of possible interrelationships among the four time series. Within the framework of our coupling coefficients, we find that carbon dioxide and temperature act to stabilize each other and methane and nitrous oxide, whereas the latter two destabilize each other and carbon dioxide and temperature. We also compute the response function for each pair of variables to assess the model performance by comparison to the data and confirm the model predictions regarding stability amongst variables. Taken together, our results are consistent with glacial pacing dominated by carbon dioxide and temperature that is modulated by terrestrial biosphere feedbacks associated with methane and nitrous oxide emissions.

Dipole-monopole criticality and chargeless half mode in an integrable gauge-coupled pseudoexciton-phonon system on a regular one-dimensional lattice.

A one-dimensional nonlinear dynamical system of gauge-coupled intrasite excitations and lattice vibrations on an infinite one-dimensional regular lattice is studied. The system as a whole is shown to be integrable in the Lax sense and it admits the exact four-component analytical solutions. Two mutually PT-conjugated symmetry broken solutions are explicitly isolated in the framework of the Darboux-Bäcklund dressing technique. Each of the obtained four-component solutions demonstrates the pronounced interplay between the interacting subsystems in the form of an essentially nonlinear superposition of two principally distinct types of traveling waves characterized by two physically distinct spatial scales and by two distinct running velocities. Depending on the relationships between the spatial scaling parameters the system can manifest itself in three qualitatively distinct dynamical regimes referred to as the monopole regime, dipole regime, and threshold regime. The threshold value of the localization parameter separating the monopole and dipole dynamical regimes is strictly established in terms of basic physical parameters. The phenomenon of dipole-monopole crossover for the spatial distribution of pseudoexcitons is shown to initiate the partial splitting between the pseudoexcitonic and vibrational subsystems in the threshold dynamical regime specified by the threshold value of the localization parameter. This partial splitting is manifested by the complete elimination of one pseudoexcitonic component accompanied by the actual conversion of another pseudoexcitonic component into a pseudoexcitonic chargeless half mode. The integrable nonlinear pseudoexciton-phonon system under study is expected to be applicable for modeling the nonlinear dynamical properties of properly designed PT-symmetric metamaterials.

One-dimensional monotone nonautonomous dynamical systems

One-dimensional monotone nonautonomous dynamical systems

Attractors in Pattern Iterations of Flat Top Tent Maps

Flat-topped one-dimensional maps have been used in the control of chaos in one-dimensional dynamical systems. In these applications, this mechanism is known as simple limiter control. In this paper, we will consider the introduction of simple limiters u in the tent map, according to a time-dependent scheme defined by a binary sequence s, the iteration pattern. We will define local and Milnor attractors in this non-autonomous context and study the dependence of their existence and coexistence on the value of the limiter u and on the pattern s. Using symbolic dynamics, we will be able to characterize the families of pairs (u,s) for which these attractors exist and coexist, as well as fully describe them. We will observe that this non-autonomous context provides a richness of behaviors that are not possible in the autonomous case.

Distribution of cycles for one-dimensional random dynamical systems

We consider an independently identically distributed random dynamical system generated by finitely many, non-uniformly expanding Markov interval maps with a finite number of branches. Assuming a topologically mixing condition and the uniqueness of equilibrium state for the associated skew product map, we establish a samplewise (quenched) almost-sure level-2 weighted equidistribution of ‘random cycles’, with respect to a natural stationary measure as the periods of the cycles tend to infinity. This result implies an analogue of Bowen's theorem on periodic orbits of topologically mixing Axiom A diffeomorphisms. We also prove another almost-sure convergence theorem, as well as an averaged (annealed) theorem that is related to semigroup actions. We apply our results to the random β-expansion of real numbers, and obtain almost-sure convergences of average digital quantities in random β-expansions of random cycles that do not follow from the application of the ergodic theorems of Birkhoff or Kakutani. Our main results are applicable to random dynamical systems generated by finitely many maps with common neutral fixed points.

Free Energy and Equilibrium States for Families of Interval Maps

We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.

Relation of stability and bifurcation properties between continuous and ultradiscrete dynamical systems via discretization with positivity: One dimensional cases

The stability and bifurcation properties of one-dimensional discrete dynamical systems with positivity, which are derived from continuous ones by tropical discretization, are studied. The discretized time interval is introduced as a bifurcation parameter in the discrete dynamical systems, and the emergence condition of an additional bifurcation, flip bifurcation, is identified. The correspondence between the discrete dynamical systems with positivity and the ultradiscrete ones derived from them is discussed. It is found that the derived ultradiscrete max-plus dynamical systems can retain the bifurcations of the original continuous ones via tropical discretization and ultradiscretization.

The peak-end rule and its dynamic realization through differential equations with maxima *

In the 1990s, after a series of experiments, the behavioural psychologist and economist Daniel Kahneman and his colleagues formulated the following peak-end evaluation rule: the remembered utility of pleasant or unpleasant episodes is accurately predicted by averaging the peak (most intense value) of instant utility (or disutility) recorded during an episode and the instant utility recorded near the end of the experience (Kahneman et al 1997 Q. J. Econ. 112 375–405). Based on this rule, we propose a mathematical model for the time evolution of the experienced utility function given by the scalar differential equation where f represents exogenous stimuli, h is the maximal duration of the experience, and are some averaging weights. In this work, we study equation and show that, for a range of parameters and a periodic sine-like term f, the dynamics of can be completely described in terms of an associated one-dimensional dynamical system generated by a piece-wise continuous map from a finite interval into itself. We illustrate our approach with two representative examples. In particular, we show that the utility u(t) (e.g. ‘happiness’, interpreted as hedonic utility) can exhibit chaotic behaviour.

Dynamics of q deformed Ricker map

We apply the deformation scheme to the classical Ricker map and obtain a q-deformed Ricker map, namely, q-Ricker map. The aim of the paper is to investigate the nonlinear dynamics, bifurcation structure, and topological entropy of q-Ricker map. In particular, we show that q-Ricker map proclaims many exciting phenomena that are remarkable in one-dimensional dynamical systems, such as the presence of coexisting attractors, physically non-observable chaos, hydra paradox, bubbling effect, and extinction. We discuss fold and flip bifurcations and further the presence of stochastically stable chaos. Finally, we show that a certain amount of deformation in the system can keep it in equilibrium; however, excessive deformation causes extinction.

Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map

The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system on some interval , so the equation induces the infinite-dimensional dynamical system on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence (as in the discrete case) but also on the internal structure of as . Assuming to be continuous, we consider the iterations of as the semigroup generated by on the space of continuous maps, and introduce the notion of a limit semigroup for in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of .

CLASSIFICATION OF ONE DIMENSIONAL DYNAMICAL SYSTEMS BY COUNTABLE STRUCTURES

AbstractWe study the complexity of the classification problem of conjugacy on dynamical systems on some compact metrizable spaces. Especially we prove that the conjugacy equivalence relation of interval dynamical systems is Borel bireducible to isomorphism equivalence relation of countable graphs. This solves a special case of Hjorth’s conjecture which states that every orbit equivalence relation induced by a continuous action of the group of all homeomorphisms of the closed unit interval is classifiable by countable structures. We also prove that conjugacy equivalence relation of Hilbert cube homeomorphisms is Borel bireducible to the universal orbit equivalence relation.

Dynamics, multistability, and crisis analysis of a sine-circle nontwist map.

We propose a one-dimensional dynamical system, the sine-circle nontwist map, that can be considered a local approximation of the standard nontwist map and an extension of the paradigmatic sine-circle map. The map depends on three parameters, exhibiting a simple mathematical form but with a rich dynamical behavior. We identify periodic, quasiperiodic, and chaotic solutions for different parameter sets with the Lyapunov exponent and Slater's theorem. From the bifurcation analysis, we determine two bifurcation lines, those that depend on just two of the control parameters, for which the bifurcation that occurs is of the saddle-node type. In order to investigate multistability, we analyze the bifurcation diagrams in the two directions of parameter variation and we observe some regions of hysteresis, representing the coexistence of different attractors. We also analyze different multistable scenarios, as single attractor, coexistence of periodic attractors, coexistence of chaotic and periodic attractors, chaotic behavior, and coexistence of different chaotic bands, by the Lyapunov exponent and the analysis of the domain occupied by the solutions. From the parameter spaces constructed, we observe the prevalence of single attractor and only chaotic behavior scenarios. The multistable scenario is, mostly, formed by different periodic attractors. Lastly, we analyze the crisis in chaotic attractors and we identify the interior and the boundary crisis. From our results, the boundary crisis plays a key role for the extinction of multistability.

IDENTIFICATION OF LINEAR DYNAMIC SYSTEMS IN THE ENVIRONMENT OF POLYNOMYAL SIGNALS

A method for the structural and parametric identification of one-dimensional linear stationary dynamic systems, represented by differential "inputoutput" constraint equations, is proposed. The method is focused on both active and passive experiments. The method is based on a polynomialrepresentation of the input and output signals of the identified dynamic system. A compact vector-matrix representation of polynomials is proposed,which makes it possible to find the forced component of the solution of linear differential equations as a result of performing simple linear algebraicoperations. The vector-matrix representation of polynomials made it possible to quite simply solve the problem of inversion of linear dynamicalsystems and the problem of compensating the measured perturbation. The issues of representing time signals in polynomial form are not considered inthis paper. Based on the obtained linear representation of a one-dimensional dynamic system, which links the parameters of the input and outputsignals with the parameters of the differential equation of the identified dynamic system mathematical model, a linear system of algebraic equations forunknown coefficients of the differential process equation is obtained. In the general case, the resulting system belongs to the class of overdeterminedsystems, and therefore its solution can be obtained by the least-square technique and is reduced to finding a pseudoinverse matrix. A block diagram ofsoftware for solving the problem of structural and parametric identification in the environment of polynomial signals is proposed. The algorithmincludes the procedure of comparing the results of numerical simulation of the identified model with the output experimental signal and correcting thestructure of the model based on the results of the comparison.

A novel second-order nonstandard finite difference method for solving one-dimensional autonomous dynamical systems

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced and analyzed. This method is based on novel non-local approximations for right-hand side functions of differential equations in combination with nonstandard denominator functions. The obtained results not only resolve the contradiction between the dynamic consistency and high-order accuracy of NSFD methods but also improve and extend some well-known results that have been published recently in Kojouharov et al. (2021); Gupta et al. (2020); Wood and Kojouharov (2015). Furthermore, as a simple but important application, we apply the constructed NSFD method for solving the logistic, sine, cubic, and Monod equations. Consequently, the NSFD schemes constructed in the earlier work (Mickens, 1999) are improved significantly. Finally, we report some numerical experiments to support and illustrate the theoretical assertions as well as advantages of the constructed NSFD method.

Wave propagation in a non-linearly elastic bar with phase transformations

There exist considerable difficulties in studying the wave propagation problems of the kind of phase transforming materials with non-monotone material responses due to the lack of uniqueness of solution for a moving phase boundary. An admissibility condition, which is named the Maxwell equal-area criterion, was established from a three-dimensional (3D) internal-variable formulation in the previous study to supplement the one-dimensional (1D) dynamical system of the bars/rods made of the special materials to determine the phase boundary uniquely. In this paper, we employ this criterion to investigate the wave propagation in the bar made of the special kind of phase transforming materials, which are associated with trilinear non-monotone stress–strain responses. The Riemann problem for different boundary states that arises from wave encounters are analyzed by the method of wave curves. Based on the construction of the Riemann solution, we explicitly study the wave propagation in a semi-infinite bar under an impulsive stress impact to explore the dynamical features of the special materials. The wave structures in the entire spatial and temporal domain for two different impact levels are obtained, as well as the state profiles. Detailed analyses and discussions of the wave patterns and wave encounters occurring during the wave propagation process are given for a deep understanding of dynamical phase transitions of the materials. These results are further compared with those obtained using the artificially adopted maximal dissipation rate criterion. The similarities and differences are discussed. The comparisons further confirmed that the special phase transforming material can be used for designing wave-absorption/impact-protection devices.

Nonlinear dynamics of an integrable gauge-coupled exciton-phonon system on a regular one-dimensional lattice

A one-dimensional nonlinear dynamical system of intra-site excitations and lattice vibrations coupled via gauge-like mechanism is studied. The system admits the semi-discrete zero-curvature representation and therefore it proves to be integrable in the Lax sense. Relaying upon an appropriately developed Darboux–Bäcklund dressing technique the explicit four-component analytical solution to the system is found and analyzed in details. Due to mutual influence between the interacting subsystems the physically meaningful solution arises as the essentially nonlinear superposition of two principally distinct types of traveling waves. The interplay between the two typical spatial scales relevant to these traveling waves causes the criticality of system’s dynamics manifested as the dipole-monopole transition in the spatial distribution of intra-site excitations.

Accuracy of a one-dimensional reduction of dynamical systems on networks.

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao etal. [Nature (London) 530, 307 (2016)0028-083610.1038/nature16948] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.