Computational Intelligence for Robust Control Algorithms of Complex Dynamic Systems with Minimum Entropy Production

The present study aims to find connections between two types of entropy production rates for continuous dynamical systems, one defined in the general dissipation context and the other one defined in a thermodynamics context. On one hand, the entropy production rate e p e_p introduced by Andrey and Ruelle in the general dynamics context is based on Shannon’s entropy in the phase space that characterizes volume contraction relating to dissipation. On the other hand, the Helmholtz-Boltzmann entropy production rate Q ˙ / T \dot {Q}/T introduced in the context of mechanical theory of thermodynamics, where Q ˙ \dot {Q} is the rate of heat generations and T T is the temperature, captures the thermodynamic entropy production relating to the mechanical energy dissipation. For certain cases of energy dissipative systems, we show that e p e_p is indeed related to Q ˙ / T \dot {Q}/T . This consistency between the general mathematical notion of dissipation and the thermodynamic entropy production in dissipative systems suggests a unifying representation of nonequilibrium phenomenon in deterministic systems based on the theory of nonlinear dynamical systems.

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach

Due to their potential applications in energy production based on waste heat, direct solar radiation or other energy sources, semiconductor materials have for years attracted the attention of theoretical and experimental researchers. The focus has been on improving the performance of thermoelectric devices through several strategies and special interest has been placed on materials with spatially inhomogeneous transport properties. Inhomogeneity can be achieved in various ways, all of them leading, to a greater or lesser extent, to an improvement of the thermoelectric performance. In this paper, general linear heat and electric charge transport processes in inhomogeneous materials are addressed. The guiding idea followed here is that there exists a relationship between inhomogeneity (structuring), minimum entropy production and performance which may be fruitfully exploited for designing more efficient thermoelectric semiconductor devices. We first show that the stationary states of such materials are minimum global entropy production states. This constitutes an extension of the validity of Prigogine’s minimum entropy principle. The heat and charge transport equations obtained within the framework of classical irreversible thermodynamics are solved to find the stationary profiles of temperature and self-consistent electric potential in a one-dimensional model of a silicon–germanium alloy subjected to an external temperature difference. This allows us to assess the effect of the spatial inhomogeneity on the thermoelectric performance. We find that, regardless of the value of the applied temperature difference, the system may efficiently operate in a regime of minimum entropy production and high efficiency.

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

In this paper, an effective method is proposed called the “Dynamic Routh’s stability Criterion (DRSC)”, which is developed using the Dynamic Pole Motion (DPM) approach (DPMA). This innovative technique extends the classical Routh’s stability criterion—traditionally limited to linear time-invariant (LTI) systems—to encompass a broader range of systems, including linear time-varying and nonlinear dynamic systems. By incorporating the behavior of pole trajectories over time, the proposed method offers a more comprehensive framework for analyzing system stability in more complex and realistic scenarios. DPM approach describes the notion of dynamic poles (DP) in a three dimensional ‘g-plane’, which is an extension to the two dimensional ‘s-plane’. The s-domain approach is based upon Laplace transform and has been used for linear time-invariant (LTI) system only. This novel g-plane framework is versatile enough to be applied to both linear and nonlinear dynamic systems. For the stability of dynamic systems, the DP must present in the left-hand side (LHS) of the g-plane. For nonlinear systems, the locations of DP are the function of system states; and these systems states are the function of initial conditions with amplitude and frequency (AnF) of input signals. Therefore, in nonlinear systems, stability is influenced not only by the initial conditions but also by the AnF of the input signals. For example, for given amplitude of the input signal, system may be unstable at low frequency; however, it may become stable at high frequency or vice-versa. These stability conditions are illustrated by several examples.

Ge and Li proposed an alternative strategy to model and synchronize two totally different nonlinear systems in the end of 2011, which provided a new version for fuzzy modeling and has been applied to several fields to simplify their modeling works and solve the mismatch problems [1]-[17]. However, the proposed model limits the number of nonlinear terms in each equation so that this model could not be used in all kinds of nonlinear dynamic systems. As a result, in this paper, a more efficient and comprehensive advanced-Ge-Li fuzzy model is given to further release the limitation and improve the effectiveness of the original one. The novel fuzzy model can be applied to all kinds of complex nonlinear systems--this is the universal strategy and only m x 2 fuzzy rules as well as two linear subsystems are needed to simulate nonlinear behaviors (m is the number of states in a nonlinear dynamic system), whatever the nonlinear terms are copious or complicated. Further, the fuzzy synchronization of two nonlinear dynamic systems with totally distinct structures can be achieved via only two sets of control gains designed through the novel fuzzy model as well as its corresponding fuzzy synchronization scheme. Two complicated dynamic systems are designed to be the illustrations, Mathieu-Van der pol system with uncertainties and Quantum-cellular neural networks nano system with uncertainties, to show the effectiveness and feasibility of the novel fuzzy model.

BackgroundThere are many differences between healthy tissue and growing tumor tissue, including metabolic, structural and thermodynamic differences. Both structural and thermodynamic differences can be used to follow the entropy differences in cancerous and normal tissue. Entropy production is a bilinear form of the rates of irreversible processes and the corresponding "generalized forces". Entropy production due to various dissipation mechanisms based on temperature differences, chemical potential gradient, chemical affinity, viscous stress and exerted force is a promising tool for calculations relating to potential targets for tumor isolation and demarcation.MethodsThe relative importance of five forms of entropy production was assessed through mathematical estimation. Using our mathematical model we demonstrated that the rate of entropy production by a cancerous cell is always higher than that of a healthy cell apart from the case of the application of external energy. Different rates of entropy production by two kinds of cells influence the direction of entropy flow between the cells. Entropy flow from a cancerous cell to a healthy cell transfers information regarding the cancerous cell and propagates its invasive action to the healthy tissues. To change the direction of entropy flow, in addition to designing certain biochemical pathways to reduce the rate of entropy production by cancerous cells, we suggest supplying external energy to the tumor area, changing the relative rate of entropy production by the two kinds of cells and leading to a higher entropy accumulation in the surrounding normal cells than in the tumorous cells.ConclusionThrough the use of mathematical models it was quantitatively demonstrated that when no external force field is applied, the rate of entropy production of cancerous cells is always higher than that of healthy cells. However, when the external energy of square wave electric pulses is applied to tissues, the rate of entropy production of normal cells may exceed that of cancerous cells. Consequently, the application of external energy to the body can reverse the direction of the entropy current. The harmful effect brought about by the entropy flow from cancerous to healthy tissue can be blocked by the reversed direction of entropy current from the irradiated normal tissue around the tumor.

The paper proposes derivative-free nonlinear Kalman Filtering for MIMO nonlinear dynamical systems. The considered nonlinear filtering scheme which is based on differential flatness theory extends the class of systems to which Kalman Filtering can be applied without the need for calculation of Jacobian matrices. To deduce if a dynamical system is differentially flat, the following should be examined: (i) the existence of the flat output, which is a variable that can be written as a function of the system's state variables (ii) the system's state variables and the input can be written as functions of the flat output and its derivatives. Nonlinear systems satisfying the differential flatness property can be written in the Brunovsky form via a transformation of their state variables and control inputs. After transforming the nonlinear system to the canonical form it is straightforward to apply the standard Kalman Filter recursion. The performance of the proposed derivative-free nonlinear filtering scheme is tested through simulation experiments on benchmark nonlinear multi-input multi-output dynamical systems, such as robotic manipulators.

30 - Minimum entropy production

Publication in the conference proceedings of EUSIPCO, Trieste, Italy, 1996

General optimization results from physics indicate that maximum efficiency of a process, in the sense of minimum overall entropy production, is achieved when the rate of entropy production is constant over time, however not in ordinary clock time but on an, in general varying, "eigen time" scale, intrinsic to the system. We identify the eigen time of a biological system with "physiological time," which generally scales with the 1/4 power of body mass, M(1/4), over a vast range of species. Since it is equally well established that metabolic rate scales as M(3/4), it follows that organisms produce entropy at the same intrinsic rate, fulfilling a necessary condition for maximum efficiency, and are all, furthermore, equally efficient on the physiological eigen time scale.

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].

BackgroundFractional-order models have become essential tools for representing nonlinear dynamical systems with memory and hereditary effects. While constant-order fractional operators have been widely studied, many real-world systems exhibit memory characteristics that evolve over time or system state. Variable-order fractional calculus addresses this limitation but introduces significant analytical and numerical challenges, particularly in nonlinear settings where stability and convergence are difficult to guarantee. Existing numerical approaches often suffer from accuracy loss and instability when fractional orders vary rapidly. MethodsThis study develops a hybrid fractional integral framework for nonlinear dynamical systems governed by variable-order operators. The methodology combines multiple variable-order fractional integrals within a unified formulation to balance short-term and long-term memory effects. A predictor–corrector numerical scheme is employed to handle nonlinearities and evolving fractional orders efficiently. Rigorous theoretical analysis is conducted to establish existence, uniqueness, convergence, and stability under standard smoothness and Lipschitz assumptions. The framework is evaluated through numerical experiments on benchmark nonlinear systems with different variable-order profiles. ResultsThe proposed hybrid framework demonstrates consistently improved numerical performance compared with constant-order and single-operator variable-order methods. Results show lower global error, higher observed convergence order, and enhanced stability under oscillatory and discontinuous fractional-order variations. The method also exhibits greater robustness in strongly nonlinear regimes, where conventional schemes experience significant error amplification. Computational efficiency is improved through faster convergence, enabling accurate solutions with fewer time steps and reduced memory overhead. ConclusionThe hybrid fractional integral framework provides a stable, accurate, and flexible approach for solving nonlinear variable-order dynamical systems. By structurally integrating complementary memory operators, the method overcomes key limitations of existing approaches and offers a reliable computational foundation for modeling complex systems with adaptive memory dynamics.