Deterministic Dynamical Systems Research Articles

A rational ansatz for the approximation of Koopman eigenfunctions

AbstractKoopman operator theory offers a basis for the systematic transformation and linearization of complex dynamical systems. We propose a method to approximate eigenfunctions of the Koopman operator for sufficiently smooth, deterministic and autonomous dynamical systems with hyperbolic fixed points in an equation‐based context. Approximations of the eigenfunctions are obtained in form of a rational ansatz whose coefficients are determined by minimizing a residual through a bi‐quadratic optimization problem. In addition, we consider an extension of the Hartman‐Grobman theorem, which was first proposed by Lan and Mezić in 2013, as a linear constraint. The implementation for a damped pendulum shows that the approach works in general, however, the optimization problem is non‐convex and thus sensible w. r. t. initial conditions, and increases proportional to the number of ansatz functions to the power of four.

Detecting causalities between strongly coupled dynamical systems

A new version of the convergent cross mapping is proposed to detect causalities from records for strongly coupled nonlinear dynamical systems, where the mutual entropy is used to measure nonlinear correlations, and the time delay stability is adopted to filter out false identifications. Calculations on various deterministic dynamic systems show that it is applicable not only to strongly coupled systems but also to non-interacting systems influenced by a common environment. Compared with the original version of convergent cross mapping, under strong couplings our proposed method has significantly higher accuracy, and is more robust to coupling strength. As a typical example, it is used to detect the causal effects between arterial blood pressure (ABP) and intracranial pressure (ICP) of patients diagnosed with traumatic brain injury (TBI). A mono-directional causality from ICP to ABP is identified.

Extension of dual equivalent linearization to analysis of deterministic dynamic systems: Part 2—multi-parameter equivalent linearization

Extension of dual equivalent linearization to analysis of deterministic dynamic systems: Part 2—multi-parameter equivalent linearization

Empirical adequacy of the time operator canonically conjugate to a Hamiltonian generating translations

To admit a canonically conjugate time operator, the Hamiltonian has to be a generator of translations (like the momentum operator generates translations in space), so its spectrum must be unbounded. But the Hamiltonian governing our world is thought to be bounded from below. Also, judging by the number of fields and parameters of the Standard Model, the Hamiltonian seems much more complicated. In this article I give examples of worlds governed by Hamiltonians generating translations. They can be expressed as a partial derivative operator just like the momentum operator, but when expressed in function of other observables they can exhibit any level of complexity. The examples include any quantum world realizing a standard ideal measurement, any quantum world containing a clock or a free massless fermion, the quantum representation of any deterministic time-reversible dynamical system without time loops, and any quantum world that cannot return to a past state. Such worlds are as sophisticated as our world, but they admit a time operator. I show that, despite having unbounded Hamiltonian, they do not decay to infinite negative energy any more than any quantum or classical world. Since two such quantum systems of the same Hilbert space dimension are unitarily equivalent even if the physical content of their observables is very different, they are concrete counterexamples to Hilbert Space Fundamentalism (HSF). Taking the observables into account removes the ambiguity of HSF and the clock ambiguity problem attributed to the Page-Wootters formalism, also caused by assuming HSF. These results provide additional motivations to restore the spacetime symmetry in the formulation of Quantum Mechanics and for the Page-Wootters formalism.

Stable large deviations for deterministic dynamical systems

We obtain large deviations for a class of non-square integrable dependent random variables in the domain of attraction of an [Formula: see text]-stable law, [Formula: see text]. This class includes ergodic sums of observables in the domain of attraction of an [Formula: see text]-stable law driven by Gibbs–Markov maps.

Time-reversibility and nonvanishing Lévy area

We give a complete description and clarification of the structure of the Lévy area correction to Itô/Stratonovich stochastic integrals arising as limits of time-reversible deterministic dynamical systems. In particular, we show that time-reversibility forces the Lévy area to vanish only in very specific situations that are easily classified. In the absence of such obstructions, we prove that there are no further restrictions on the Lévy area and that it is typically nonvanishing and far from negligible.

An information field theory approach to Bayesian state and parameter estimation in dynamical systems

Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. When the dynamics are discrete and stochastic, one may employ powerful techniques such as Kalman, particle, or variational filters. Practitioners commonly apply these methods to continuous-time, deterministic dynamical systems after discretizing the dynamics and introducing fictitious transition probabilities. However, approaches based on time-discretization suffer from the curse of dimensionality since the number of random variables grows linearly with the number of time-steps. Furthermore, the introduction of fictitious transition probabilities is an unsatisfactory solution because it increases the number of model parameters and may lead to inference bias. To address these drawbacks, the objective of this paper is to develop a scalable Bayesian approach to state and parameter estimation suitable for continuous-time, deterministic dynamical systems. Our methodology builds upon information field theory. Specifically, we parameterize the dynamical system state path using a linear basis. Then, we construct a physics-informed prior probability measure for the coefficients of the linear basis such that functions that satisfy the physics are more likely. This prior allows us to quantify model form errors. We connect the system's response to observations through a probabilistic model of the measurement process. The joint posterior over the system responses and all parameters is given by Bayes' rule. To approximate the intractable posterior, we develop a stochastic variational inference algorithm. In summary, the developed methodology offers a powerful framework for Bayesian estimation in dynamical systems.

Testing dynamic correlations and nonlinearity in bivariate time series through information measures and surrogate data analysis.

The increasing availability of time series data depicting the evolution of physical system properties has prompted the development of methods focused on extracting insights into the system behavior over time, discerning whether it stems from deterministic or stochastic dynamical systems. Surrogate data testing plays a crucial role in this process by facilitating robust statistical assessments. This ensures that the observed results are not mere occurrences by chance, but genuinely reflect the inherent characteristics of the underlying system. The initial process involves formulating a null hypothesis, which is tested using surrogate data in cases where assumptions about the underlying distributions are absent. A discriminating statistic is then computed for both the original data and each surrogate data set. Significantly deviating values between the original data and the surrogate data ensemble lead to the rejection of the null hypothesis. In this work, we present various surrogate methods designed to assess specific statistical properties in random processes. Specifically, we introduce methods for evaluating the presence of autodependencies and nonlinear dynamics within individual processes, using Information Storage as a discriminating statistic. Additionally, methods are introduced for detecting coupling and nonlinearities in bivariate processes, employing the Mutual Information Rate for this purpose. The surrogate methods introduced are first tested through simulations involving univariate and bivariate processes exhibiting both linear and nonlinear dynamics. Then, they are applied to physiological time series of Heart Period (RR intervals) and respiratory flow (RESP) variability measured during spontaneous and paced breathing. Simulations demonstrated that the proposed methods effectively identify essential dynamical features of stochastic systems. The real data application showed that paced breathing, at low breathing rate, increases the predictability of the individual dynamics of RR and RESP and dampens nonlinearity in their coupled dynamics.

A New Model Making the Transition From Oil to Solar Energy

Today, the use of renewable energy sources such as sun, wind and hot water to generate electricity is increasingly becoming a global priority. In this paper, we propose a deterministic dynamical system reporting the transition from oil to solar energy, in which we then add the randomness of this phenomenon. First, we study the positivity of the deterministic oil-solar model and show that there is a maximum solution of the deterministic model and that the system is controllable at any time T > 0 from an initial point. We also prove the existence of an equilibrium state, study its nature and prove that there is no limit cycle between the quantity of oil available and the solar production capacity installed. Secondly, we formulate the stochastic solar energy model from the deterministic model, for which we proved the existence of a unique solution. Finally, using a model extracted from the model performed and examining the transition from the amount of available oil to the installed solar generation capacity, we associate a linear optimal control problem, from which we prove that the optimal control components are of bang-bang type, and we characterize them.

Wigner- and Marchenko–Pastur-Type Limit Theorems for Jacobi Processes

AbstractWe study Jacobi processes $$(X_{t})_{t\ge 0}$$ ( X t ) t ≥ 0 on $$[-1,1]^N$$ [ - 1 , 1 ] N and $$[1,\infty [^N$$ [ 1 , ∞ [ N which are motivated by the Heckman–Opdam theory and associated integrable particle systems. These processes depend on three positive parameters and degenerate in the freezing limit to solutions of deterministic dynamical systems. In the compact case, these models tend for $$t\rightarrow \infty $$ t → ∞ to the distributions of the $$\beta $$ β -Jacobi ensembles and, in the freezing case, to vectors consisting of ordered zeros of one-dimensional Jacobi polynomials. We derive almost sure analogues of Wigner’s semicircle and Marchenko–Pastur limit laws for $$N\rightarrow \infty $$ N → ∞ for the empirical distributions of the N particles on some local scale. We there allow for arbitrary initial conditions, which enter the limiting distributions via free convolutions. These results generalize corresponding stationary limit results in the compact case for $$\beta $$ β -Jacobi ensembles and, in the deterministic case, for the empirical distributions of the ordered zeros of Jacobi polynomials. The results are also related to free limit theorems for multivariate Bessel processes, $$\beta $$ β -Hermite and $$\beta $$ β -Laguerre ensembles, and the asymptotic empirical distributions of the zeros of Hermite and Laguerre polynomials for $$N\rightarrow \infty $$ N → ∞ .

Приближенный синтез оптимальных детерминированных систем управления с неполной обратной связью на основе достаточных условий ε-оптимальности

<p>The problem of optimal control of deterministic dynamical systems in the absence of information about a part of the coordinates of the state vector is considered. Sufficient ε-optimality conditions based on the principle of expansion are formulated and proved. An algorithm is proposed for finding an a priori estimate of the proximity of the synthesized control law with incomplete feedback to the optimal one for a given set of initial states. The solution of the model example is given.</p>

A Simple Model for Targeting Industrial Investments with Subsidies and Taxes

We consider an investor, whose capital is divided into an industrial investment xt and cash yt, and satisfy a nonlinear deterministic dynamical system. The investor fixes fractions of capital to be invested, withdrawn, and consumed, and also the production factor parameter. The government fixes a subsidy fraction for industrial investments and a tax fraction for the capital outflow. We study a Stackelberg game, corresponding to the asymptotically stable equilibrium (x∗,y∗) of the mentioned dynamical system. In this game, the government (the leader) uses subsidies to make incentives for the investor (the follower) to maintain the desired level of x∗, and uses taxes to achieve this with the minimal cost. The investor’s aim is to maximize the difference between the consumption and the price of the production factor at equilibrium. We present an explicit analytical solution of the specified Stackelberg game. Based on this solution, we introduce the notion of a fair industrial investment level, which is costless for the government, and show that it can produce realistic results using a case study of water production in Lahore.

Detecting the Non‐Separable Causality in Soil Moisture‐Precipitation Coupling With Convergent Cross‐Mapping

AbstractSoil moisture‐Precipitation (SM‐P) coupling, especially the causality from SM to P (SM → P), is an important and highly debated topic. The causal inference from observational data provides a statistical approach for this issue, where the experimental research is infeasible. Various causal inference methods exist, each assuming distinct underlying systems for the targeted variables: pure stochastic or deterministic dynamic system (DDS), which means that these methods detect different types of causality: separable or non‐separable. Acknowledging the inherent deterministic dynamic nature of SM‐P coupling, this study employs a DDS‐based method: Convergent Cross‐mapping (CCM), to detect their non‐separable causality. Centering around SM‐P coupling, we also detect the causality between SMs in shallow and deeper layers, and the causality between evapotranspiration (ET) and SMs in all layers. Notably, before applying CCM, a preconditional procedure is required: verifying the DDS nature of targeted variables P, SM, and ET. Key findings of this research include: (a) In SM‐P coupling, only SMs in shallow layers but not deeper layers could render causalities toward P, while P renders causalities toward SMs in all layers. (b) The time‐delay of causality from SM1 to P in spring/summer is around 4–6 days, and that from P to SM1 is around 2–4 days. (c) Inside SMs, shallow SMs have obvious causalities downwards to deeper SMs, but deeper SMs seem harder to render causalities upwards to shallow SMs, explaining why they hardly render causalities to P. In summary, this study furnishes an indispensable DDS‐based complement to stochastic‐based methodologies for SM‐P coupling.

Stability of Traffic Equilibria in a Day-to-Day Dynamic Model of Route Choice and Adaptive Signal Control

Adaptive traffic signal control has the potential to promote the efficient use of road intersections, thus contributing to the effectiveness of urban traffic management schemes. However, the reaction of drivers to repeatedly updated signal settings and the ensuing route choice dynamics may trigger the emergence of various kinds of network instability. In this study, the joint evolution of traffic flows and adaptive signal settings in a road network is investigated at the level of day-to-day dynamics with an explicit focus on the stability issue. We show how a Logit form signal control policy can be used, in interaction with route choice, to counter the emergence of instabilities possibly arising as a consequence of various behavioral factors and network conditions. After providing a general formulation of the model as a discrete time, deterministic nonlinear dynamical system, an explicit analysis of fixed-point stability is carried out for a simple network. Numerical results obtained from the implementation of the model on two example networks are presented in order to support the analytical findings of this study. We conclude that, in an integrated traffic management and information system, a properly calibrated adaptive signal control policy has the potential to offset the destabilizing effect of highly accurate driver information supplied by navigational aids. Our findings also suggest that the Logit-like control policy performs better than the Equisaturation signal setting method, in terms of average intersection delay at equilibrium, for all levels of driver information and travel demand tested in the experiment.

Method for phase space reconstruction to estimate the short-term future behavior of pressure signals in pipelines

In this study, we propose a method based on phase space reconstruction to estimate the short-term future behavior of pressure signals in pipelines. The pressure time series data were obtained from an IoT experimental model conducted in the laboratory. The proposed hydraulic system demonstrated the presence of traces of weak chaos in the time series of the pressure signal. Fractal dimension analysis revealed a complex fractal structure in the data, indicating the existence of nonlinear dynamics. Similarly, Lyapunov coefficients, divergent trajectories, and autocorrelation analysis confirmed the presence of weak chaos in the time series. The results demonstrated the existence of apparently chaotic patterns that follow the theory proposed by Kolmogorov for deterministic dynamic systems that exhibit apparently random behaviors. Phase space reconstruction allowed us to show the dynamic characteristics of the signal so that short-term predictions were stable. Finally, the study of strange attractors in pipeline pressure time series can have significant contributions to anomaly detection.•A methodology is proposed for the reconstruction of the phase space to estimate the short-term future behavior of pressure signals in pipelines in real time.•The analysis of the proposed hydraulic system revealed some indications of weak chaos in the time series of the pressure signal obtained experimentally.•The methodology implemented and the results of this study showed that the short-term predictions were very accurate and consistent; Chaotic patterns were also identified that support the theory proposed by Kolmogorov.

Dual symplectic classical circuits: An exactly solvable model of many-body chaos

We propose a general exact method of calculating dynamical correlation functions in dual symplectic brick-wall circuits in one dimension. These are deterministic classical many-body dynamical systems which can be interpreted in terms of symplectic dynamics in two orthogonal (time and space) directions. In close analogy with quantum dual-unitary circuits, we prove that two-point dynamical correlation functions are non-vanishing only along the edges of the light cones. The dynamical correlations are exactly computable in terms of a one-site Markov transfer operator, which is generally of infinite dimensionality. We test our theory in a specific family of dual-symplectic circuits, describing the dynamics of a classical Floquet spin chain. Remarkably, expressing these models in the form of a composition of rotations leads to a transfer operator with a block diagonal form in the basis of spherical harmonics. This allows us to obtain analytical predictions for simple local observables. We demonstrate the validity of our theory by comparison with Monte Carlo simulations, displaying excellent agreement with the latter for a choice of observables.

Fourier series-based approximation of time-varying parameters in ordinary differential equations

Many real-world systems modeled using differential equations involve unknown or uncertain parameters. Standard approaches to address parameter estimation inverse problems in this setting typically focus on estimating constants; yet some unobservable system parameters may vary with time without known evolution models. In this work, we propose a novel approximation method inspired by the Fourier series to estimate time-varying parameters (TVPs) in deterministic dynamical systems modeled with ordinary differential equations. Using ensemble Kalman filtering in conjunction with Fourier series-based approximation models, we detail two possible implementation schemes for sequentially updating the time-varying parameter estimates given noisy observations of the system states. We demonstrate the capabilities of the proposed approach in estimating periodic parameters, both when the period is known and unknown, as well as non-periodic TVPs of different forms with several computed examples using a forced harmonic oscillator. Results emphasize the importance of the frequencies and number of approximation model terms on the time-varying parameter estimates and corresponding dynamical system predictions.

Transition to anomalous dynamics in a simple random map.

The famous doubling map (or dyadic transformation) is perhaps the simplest deterministic dynamical system exhibiting chaotic dynamics. It is a piecewise linear time-discrete map on the unit interval with a uniform slope larger than one, hence expanding, with a positive Lyapunov exponent and a uniform invariant density. If the slope is less than one, the map becomes contracting, the Lyapunov exponent is negative, and the density trivially collapses onto a fixed point. Sampling from these two different types of maps at each time step by randomly selecting the expanding one with probability p, and the contracting one with probability 1-p, gives a prototype of a random dynamical system. Here, we calculate the invariant density of this simple random map, as well as its position autocorrelation function, analytically and numerically under variation of p. We find that the map exhibits a non-trivial transition from fully chaotic to completely regular dynamics by generating a long-time anomalous dynamics at a critical sampling probability pc, defined by a zero Lyapunov exponent. This anomalous dynamics is characterized by an infinite invariant density, weak ergodicity breaking, and power-law correlation decay.

Hydrological records can be used to reconstruct the resilience of watersheds to climatic extremes

Hydrologic resilience modeling is used in public watershed management to assess watershed ability to supply life-supporting ecoservices under extreme climatic and environmental conditions. Literature surveys criticize resilience models for failing to capture watershed dynamics and undergo adequate testing. Both shortcomings compromise their ability to provide management options reliably protecting water security under real-world conditions. We formulate an empirical protocol to establish real-world correspondence. The protocol applies empirical nonlinear dynamics to reconstruct hydrologic dynamics from watershed records, and analyze the response of reconstructed dynamics to extreme regional climatic conditions. We devise an AI-based early-warning system to forecast (out-of-sample) reconstructed hydrologic resilience dynamics. Application to the La Tejería (Spain) experimental watershed finds it to be a low dimensional nonlinear deterministic dynamic system responding to internal stressors by irregularly oscillating along a watershed attractor. Reconstructed and forecasted hydrologic resilience behavior faithfully captures monthly wet-cold/dry-warm weather patterns characterizing the Mediterranean region.

Generating pseudo-random numbers with a Brownian system

We use a deterministic dynamical system that simulates a Brownian movement over time to produce pseudo-random sequences of binary numbers. We show that the used Brownian system has two positive Lyapunov exponents, this characteristic is different of a chaotic system that has only one maximum positive Lyapunov exponent. The implementation of the random number generator uses fixed point arithmetic with numbers with 5 bits for the integer part and 58 bits for the fractional part. The eight least significant bits for each variable of the 3rd order Brownian system are concatenated to form the sequences of random bits. The generated sequences pass all the NIST and TestU01 statistical randomness tests. Furthermore, a hardware design in FPGA of the proposed pseudo-random generator is presented with a throughput equal to 1085.16 Mb/s. For the best of our knowledge, this is the first time a Brownian system is used as the core of a PRNG.