Nonlinear Discrete Dynamical Systems Research Articles

Nonlinear causal network learning via Granger causality based on extreme support vector regression.

For complex networked systems, based on the consideration of nonlinearity and causality, a novel general method of nonlinear causal network learning, termed extreme support vector regression Granger causality (ESVRGC), is proposed. The nonuniform time-delayed influence of the driving nodes on the target node is particularly considered. Then, the restricted model and the unrestricted model of Granger causality are, respectively, formulated based on extreme support vector regression, which uses the selected time-delayed components of system variables as the inputs of kernel functions. The nonlinear conditional Granger causality index is finally calculated to confirm the strength of a causal interaction. Generally, based on the simulation of a nonlinear vector autoregressive model and nonlinear discrete time-delayed dynamic systems, ESVRGC demonstrates better performance than other popular methods. Also, the validity and robustness of ESVRGC are also verified by the different cases of network types, sample sizes, noise intensities, and coupling strengths. Finally, the superiority of ESVRGC is successful verified by the experimental study on real benchmark datasets.

On–Off Intermittency and Long-Term Reactivity in a Host–Parasitoid Model with a Deterministic Driver

Bursting behaviors, driven by environmental variability, can substantially influence ecosystem services and functions and have the potential to cause abrupt population breakouts in host–parasitoid systems. We explore the impact of environment on the host–parasitoid interaction by investigating separately the effect of grazing-dependent habitat variation on the host density and the effect of environmental fluctuations on the average host population growth rate. We hence focus on the discrete host–parasitoid Beddington–Free–Lawton model and show that a more comprehensive mathematical study of the dynamics behind the onset of on–off intermittency in host–parasitoid systems may be achieved by considering a deterministic, chaotic system that represents the dynamics of the environment. To this aim, some of the key model parameters are allowed to vary in time according to an evolution law that can exhibit chaotic behavior. Fixed points and stability properties of the resulting 3D nonlinear discrete dynamical system are investigated and on–off intermittency is found to emerge strictly above the blowout bifurcation threshold. We show, however, that, in some cases, this phenomenon can also emerge in the sub-threshold. We hence introduce the novel concept of long-term reactivity and show that it can be considered as a necessary condition for the onset of on–off intermittency. Investigations in the time-dependent regimes and kurtosis maps are provided to support the above results. Our study also suggests how important it is to carefully monitor environmental variability caused by random fluctuations in natural factors or by anthropogenic disturbances in order to minimize its effects on throphic interactions and protect the potential function of parasitoids as biological control agents.

Anomaly detection and prediction evaluation for discrete nonlinear dynamical systems

Anomalies in dynamical systems mostly occur as deviations between measurement and prediction. Current anomaly detection methods in multivariate time series often require prior clustering, training data, or cannot distinguish local and global anomalies. Furthermore, no generalized metric exists to evaluate and compare different prediction functions regarding their amount of anomalous behavior. We propose a novel methodology to detect local and global anomalies in time series data of dynamical systems. For this purpose, a theoretical density distribution is derived assuming that only noise conceals the time series. If the theoretical and the empirical density distribution yield significantly different entropies, an anomaly is assumed. For a local anomaly detection, the Mahalanobis distance using the theoretical noise distribution’s covariance is applied to evaluate sequences of predictions and measurements. In addition, the Wasserstein metric enables a comparison of predictions using the distance between the noise and empirical distribution as a measure for selecting the best prediction function. The proposed method performs well on nonlinear time series such as logistic growth and enables a useful selection of a prediction model for satellite orbits. Thus, the proposed method improves anomaly detection in time series and model selection for nonlinear systems.

System identification using autoregressive Bayesian neural networks with nonparametric noise models

System identification is of special interest in science and engineering. This article is concerned with a system identification problem arising in stochastic dynamic systems, where the aim is to estimate the parameters of a system along with its unknown noise processes. In particular, we propose a Bayesian nonparametric approach for system identification in discrete time nonlinear random dynamical systems assuming only the order of the Markov process is known. The proposed method replaces the assumption of Gaussian distributed error components with a flexible family of probability density functions based on Bayesian nonparametric priors. Additionally, the functional form of the system is estimated by leveraging Bayesian neural networks, which leads to flexible uncertainty quantification. Hamiltonian Monte Carlo sampler within a Gibbs sampler for posterior inference is proposed and its effectiveness is illustrated in real time series.

Investigation of complex dynamics and chaos control of the duopoly supply chain under the mixed carbon policy

Against the background of the mixed carbon policy (carbon tax‑carbon trading) implemented in China, the authors analyze the complexity of a series of competition decisions between supply chains through constructing Cournot(Duopoly)game model between supply chains. The dynamics of the proposed game model and the time evolution of the supply chains are described by a nonlinear discrete dynamical system. The equilibrium point of the discrete dynamical system is obtained and its local stability is discussed. The stability of the equilibrium point is further discussed by numerical simulation, which is affected by the speed of adjusting parameters and some other cost parameters. Finally, we apply the delayed feedback control method to suppress and overcome the chaos of the discrete dynamical system. It can effectively promote the discrete dynamical system to return to the stable behavior. It is hoped that the current research can provide a theoretical basis for the competitive decision-making of supply chain node enterprises.

Data-based modeling and identification for general nonlinear dynamical systems by the multidimensional Taylor network

PurposeThe present study is intended to develop an effective approach to the real-time modeling of general dynamic nonlinear systems based on the multidimensional Taylor network (MTN).Design/methodology/approachThe authors present a detailed explanation for modeling the general discrete nonlinear dynamic system by the MTN. The weight coefficients of the network can be obtained by sampling data learning. Specifically, the least square (LS) method is adopted herein due to its desirable real-time performance and robustness.FindingsCompared with the existing mainstream nonlinear time series analysis methods, the least square method-based multidimensional Taylor network (LSMTN) features its more desirable prediction accuracy and real-time performance. Model metric results confirm the satisfaction of modeling and identification for the generalized nonlinear system. In addition, the MTN is of simpler structure and lower computational complexity than neural networks.Research limitations/implicationsOnce models of general nonlinear dynamical systems are formulated based on MTNs and their weight coefficients are identified using the data from the systems of ecosystems, society, organizations, businesses or human behavior, the forecasting, optimizing and controlling of the systems can be further studied by means of the MTN analytical models.Practical implicationsMTNs can be used as controllers, identifiers, filters, predictors, compensators and equation solvers (solving nonlinear differential equations or approximating nonlinear functions) of the systems of ecosystems, society, organizations, businesses or human behavior.Social implicationsThe operating efficiency and benefits of social systems can be prominently enhanced, and their operating costs can be significantly reduced.Originality/valueNonlinear systems are typically impacted by a variety of factors, which makes it a challenge to build correct mathematical models for various tasks. As a result, existing modeling approaches necessitate a large number of limitations as preconditions, severely limiting their applicability. The proposed MTN methodology is believed to contribute much to the data-based modeling and identification of the general nonlinear dynamical system with no need for its prior knowledge.

Stability Analysis of the Nabla Distributed-Order Nonlinear Systems

The stability of the nabla discrete distributed-order nonlinear dynamic systems is investigated in this paper. Firstly, a sufficient condition for the asymptotic stability of the nabla discrete distributed-order nonlinear systems is proposed based on Lyapunov direct method. In addition, some properties of the nabla distributed-order operators are derived. Based on these properties, a simpler criterion is provided to determine the stability of such systems. Finally, two examples are given to illustrate the validity of these results.

A Dynamic System Approach to Spiking Second Order Memristor Networks

Second order memristors are two terminal devices that present a conductance depending on two orders of variables, namely the geometric parameters and the internal temperature. They have shown to be able to mimic some specific features of neuron synapses, specifically Spike-Timing-Dependent-Plasticity (STDP), and consequently to be good candidates for neuromorphic computing. In particular, memristor crossbar structures appear to be suitable for implementing locally competitive algorithms and for tackling classification problems by exploiting temporal learning techniques. On the other hand, neuromorphic studies and experiments have revealed the existence of different kinds of plasticity and have shown the effect of calcium concentration on synaptic changes. Computational studies have investigated the behavior of spiking networks in the context of supervised, unsupervised, and reinforcement learning. In this paper, we first derive a simplified, almost analytical, model of a second-order memristor, only involving two variables, the mem-conductance, and the temperature, directly attributable to the synaptic efficacy and to the calcium concentration. Then we study in detail the response of a single memristive synapse to the most relevant plasticity models, including cycles of spike pairs, triplets, and quadruplets at different frequencies. Finally, we accurately characterize memristor spiking networks as discrete nonlinear dynamic systems, with mem-conductances as state variables and pre and postsynaptic spikes as inputs and outputs, respectively. The result shows that the model developed in this manuscript can explain and accurately reproduce a significant portion of observed synaptic behaviors, including those not captured by classical spike pair-based STDP models. Furthermore, under such an approach, the global dynamic behavior of memristor networks and the related learning mechanisms can be deeply analyzed by employing advanced nonlinear dynamic techniques.

SECURITY APPLICATION OF NEURAL NETWORKS UNDER THE INSPECTION OF NONLINEAR DYNAMIC SYSTEMS

Based on the improved BP neural network, this paper establishes an adaptive online controlling model and adopts the model to optimize the controlling accuracies in discrete nonlinear dynamic systems and inverted pendulum systems. To avoid the local minimum problem of the BP neural network’s objective function in the training process, this paper proposes a neural network training method based on the quasi-Newton method (BFGS) optimization algorithm. Compared with other control methods, the neural network-based inverted pendulum control method proposed in this paper has higher control accuracy. Through the control simulation of its power system uses a discrete control method and the control of the inverted pendulum model system, this paper verifies the validity and good control has significantly improved the control method.

BASIC PRINCIPLES FOR CONSTRUCTING MIXING FUNCTIONS BASED ON THE SIMPLEST LINEAR AND NONLINEAR MAPPINGS

This article is dedicated to exploring the possibility of using the simplest principles of nonlinear discrete dynamic systems theory in computer cryptography, which are characterized by their chaotic behavior. The main problem of using chaotic systems in computer calculations is that the number of possible states in a computer is finite. Therefore, computer models of chaos are only an approximation of the true chaotic behavior, and each trajectory of the approximated system is periodic. From a mathematical point of view, encryption in information systems involves transforming the space of finite messages, which is similar to the phase space in the theory of dynamical systems. The mixing function specifies such encryption. The main requirements for the mixing function are the absence of collisions, i.e., bijectivity of the mapping, good diffusion properties, and, in addition, the inverse transformation should not be more complicated than the direct one. The article demonstrates that it is possible to utilize the diffusion properties of nonlinear dynamical systems in spaces with a finite number of states by using the simplest nonlinear mapping, Tent. To enhance the diffusion properties, a superposition of the nonlinear Tent map and the linear permutation map (in the more general case of the Hill map) was used. The main advantages of the constructed functions are their simplicity of implementation, speed of calculations in mixing problems, and strong cryptographic persistence. Correlation analysis, sensitivity analysis, and analysis of the lengths of cycle periods that divide the space into non-overlapping subsets have been conducted for these functions. As a result, the expected good diffusion properties of these mixing functions are confirmed. The possibility of applying these functions to image encryption is demonstrated.

A New Friedman’s Model for Evolutionary Game Problem

The term game in game theory means a problem, where some of the people or groups (players) share a set of rules and regulations that create the conditions and events that make up the beginning of the game. For instance, in the trade market, the buyers and sellers of some commodities interact in a random way. The reputation of sellers effects on increasing of selling. e.g., honest sellers are more attractive than cheaters for the buyers and the buyers can examine the products or unexamined. In this paper, a non-linear discrete dynamical system of Friedman model was considered. Also, we proposed a new model of interaction between these two populations (buyers and sellers). investigate its limit run behavior where we found the limit converges to a fixed point (0,0) i.e., the sellers will always cheat and the sellers will not examine according to Friedman’s model which is denoted by the fixed point (0.0).

Converging outer approximations to global attractors using semidefinite programming

This paper develops a method for obtaining guaranteed outer approximations for global attractors of continuous and discrete time nonlinear dynamical systems. The method is based on a hierarchy of semidefinite programming problems of increasing size with guaranteed convergence to the global attractor. The approach taken follows an established line of reasoning, where we first characterize the global attractor via an infinite dimensional linear programming problem (LP) in the space of Borel measures. The dual to this LP is in the space of continuous functions and its feasible solutions provide guaranteed outer approximations to the global attractor. For systems with polynomial dynamics, a hierarchy of finite-dimensional sum-of-squares tightenings of the dual LP provides a sequence of outer approximations to the global attractor with guaranteed convergence in the sense of volume discrepancy tending to zero. The method is very simple to use and based purely on convex optimization. Numerical examples with the code available online demonstrate the method.

Dynamic Formation of Customers’ Quality Expectations

In this paper, based on well-established behavioral theories, we propose a new model for dynamic formation of customer’ quality expectations. The model is described by a one-dimensional non-linear discrete dynamical system which depends on several parameters with links to behavioral theories. Applying stability analysis, we investigate the dynamic properties of the non-linear system. We find those domains in the parameter space in which the equilibrium point of the system (the fixed point) and the periodic orbits are stable and in which they are unstable. These domains correspond to the cases customers will converge to a specific expectation of the offering, while in other cases their expectations will oscillate between a small number of points periodically. These results can help identify target segments for companies and pave the way for interventions that foster more accurate customer expectations.

Robust analysis for data-driven model predictive control

Here the idea of data driven is introduced in model predictive control to establish our proposed data-driven model predictive control. Considering one first-order discrete time nonlinear dynamical system, the main essence of data driven means the actual output value in cost function for model predictive control is identified through input–output observed data in case of unknown but bounded noise and martingale difference sequence. After substituting the identified actual output in cost function, the total cost function in model predictive control is reformulated as its standard form, i.e. one quadratic program problem with input and output constraints. Then semidefinite relaxation scheme is used to derive a lower bound for its optimal value, and the robust counterpart of an uncertain quadratic program is reduced to one conic quadratic problem. The above semidefinite relaxation scheme and conic quadratic problem correspond to the similar robust analysis based on convex optimization theory. Finally, one simulation example is used to prove the efficiency of our proposed theory.

Efficiency analysis of polynomial filtering algorithms in navigation data processing for a class of nonlinear discrete dynamical systems

A polynomial filtering algorithm is designed for estimating a Markov sequence based on linear measurements. The feature of the estimation problem is that the Markov sequence is described by a nonlinear shaping filter, which is a second-order polynomial with respect to the state vector components. The algorithm efficiency is illustrated by three examples of navigation data processing. It is shown that the polynomial filter provides accuracy close to the best potential one calculated using the particle filter. At the same time, the amount of computation required to implement this algorithm is much smaller than that for the particle filter. In addition, the polynomial filter provides a consistent calculated accuracy characteristic.

PREDICTIVE CONTROL METHODS IN TASKS OF SEARCHING SADDLE POINTS

The article presents new methods for searching critical points of a function of several variables, including saddle points. Such problems are found in various fields of theoretical and practical science, for example, saddle-point construction lens design, machine and deep learning, problems of convex optimization and nonlinear programming (necessary and sufficient conditions for the solution are formulated using saddle points of the Lagrange function and proved in the Kuhn-Tucker theorem. When training neural networks, it is necessary to repeat the training process on large clusters and check the network's trainability at different loss functions and different network depth. Which means that thousands of new calculations are run, where each time the loss function is optimized on large amounts of data. So any acceleration in the process of finding critical points is a major advantage and saves computing resources. Many modern methods of searching saddle points are based on calculating the Hessian matrix, inverting this matrix, the scalar product of the gradient vector and the current vector, finding the full Lagrangian, etc. However, all these operations are computationally “expensive” and it would make sense to bypass such complex calculations. The idea of modifying the standard gradient methods used in the article is to apply fixed-point search schemes for nonlinear discrete dynamical systems for gradient descent problems. It is assumed that these fixed points correspond to unstable equilibrium positions, and there are large units among the multipliers of each equilibrium position. The averaged predictive control methods are used. Results of numerical modeling and visualization are presented in the form of two tables, which indicate basins of attraction for each critical point in each scheme, and statistical data by the convergence rates.

Toward a Comprehensive Theory for Stability Regions of a Class of Nonlinear Discrete Dynamical Systems

A comprehensive theory for the stability boundaries and the stability regions of a general class of nonlinear discrete dynamical systems is developed in this article. This general class of systems is modeled by diffeomorphisms and admits as limit sets only fixed points and periodic orbits. Topological and dynamical characterizations of stability boundaries are developed. Necessary and sufficient conditions for fixed points and periodic orbits to lie on the stability boundary are derived. Numerical examples, including applications to associative neural networks illustrating the theoretical developments, are presented.

On bifurcations and local stability in 1-D nonlinear discrete dynamical systems

In this paper, a theory of bifurcations and local stability of fixed-points (or period-1 solutions) in one-dimensional nonlinear discrete dynamical systems is presented. The linearized discrete dynamical systems are discussed first, and the higher-order singularity and monotonic and oscillatory stability of fixed-points for one-dimensional nonlinear discrete dynamical systems are presented. The monotonic and oscillatory bifurcations of fixed-points (period-1 solutions) are presented. A few special examples in 1-dimensional maps are presented for a better understanding of the general theory for the stability and bifurcation of nonlinear discrete dynamical systems. Global analysis of period-2 motions for the sampled nonlinear discrete dynamical systems are carried out, and global illustrations of period-1 to period-2 solutions in the sampled nonlinear discrete dynamical systems are given.

Application of bio-inspired methods for optimal open loop discrete time deterministic control problems

The article proposed the novel Sparrow colony optimization (SCO) algorithm to solve the optimal open loop control problem for nonlinear discrete deterministic dynamic systems described by difference equations. Numerical algorithm and software in Python programming language are implemented. Model examples demonstrating the effectiveness of the developed software are solved. Comparison of the considered algorithm results with a Dragonfly optimization algorithm at the solution of model problems is made. The recommended parameters values of the proposed algorithm are suggested.

Control of continuous dynamical systems modeling physiological states

• The predictive control to remove periodic orbits and higher order orbits when the dynamic evolves in a continuous time. • The generalization to continuous-time systems of the results obtained on the suppression of periodic dynamics in multidimensional nonlinear discrete-time systems. • The control of some undesirable physiological states in the brain as well as in the heart. • The simulation results are based on the Rossler and the FitzHugh-Nagumo models. The aim of this work is a generalization to continuous dynamical systems of the results obtained for nonlinear discrete dynamical systems, i.e. the suppression of stable periodic dynamics, especially those corresponding to certain undesirable physiological states in the brain as well as in the heart. In most cases, complex pathologies are modeled by nonlinear dynamical systems with chaotic behavior and some pathological states correspond to periodic behaviors. We propose a new strategy using predictive control to suppress periodic and higher-order orbits when the dynamics evolves in continuous time. The feasibility and efficiency of our approach is illustrated using two models, the Rössler and FitzHugh-Nagumo models.