Dynamic visual motion estimation from subspace constraints

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

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.

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

In this study, nonlinear dynamic systems are identified by using artificial bee colony (ABC) algorithm and adaptive neuro fuzzy inference system (ANFIS). ABC algorithm is used in training and updating of ANFIS. The most appropriate model is formed by optimizing the antecedent and conclusion parameters that are found in the structure of ANFIS. The dynamic systems that consist of one input and one output (SISO) are used for the identification of nonlinear dynamic systems. The obtained results are compared with fuzzy neural network, neural network and ANFIS-based methods such as RSONFIN, DFNN, RSEFNN-LF, WRFNN and RFNN. The simulation results show that the proposed method is successful in the identification of considered nonlinear dynamic systems.

This article presents a theoretical framework for the identification of nonlinear dynamic MISO systems with orthonormal base function models on fundamental basics of the Volterra theory. In the past the Volterra theory was used for the identification of nonlinear dynamic SISO systems (e.g. Hammerstein and Wiener models). In principle it is possible to extend these approaches to systems with more than one input (i.e. MISO systems). In former times this failed due to a lack of computational performance. In this paper an approach is presented which allows the identification of arbitrary coupled Hammerstein and Wiener models with multiple inputs. Some fundamental considerations for the identification of MISO systems based on arbitrary coupled Hammerstein models are made and extended to MISO systems based on arbitrary coupled Wiener models. This extended Volterra theory results in equations where the unknown parameters can be separated of the input values in a linear manner. In order to approximate the truncated impulse responses of the linear dynamic systems and to reduce the number of unknown parameters orthonormal base functions (OBFs) are introduced. As an example the proposed identification method is applied to a MISO system based on forward and backward coupled Hammerstein and Wiener models.

In this paper, we consider an observer design by using a formal linearization based on Fourier expansion for nonlinear dynamic and measurement systems. A non-linear dynamic system is given by a nonlinear ordinary differential equation, and a measurement sysetm is done by a nonlinear equation. Defining a linearization function which consists of the trigonometric functions considered up to the higher-order, a nonlinear dynamic system is transformed into an augmented linear one with respect to this linearization function by using Fourier expansion. Introducing an augmented measurement vector which consists of polynomials of measurement data, a measurement equation is transformed into an augmented linear one with respect to the linearization function in the same way. To these augmented linearized systems, a linear estimation theory is applied to design a new non-linear observer.

Nonlinear dynamic systems are characterized with uncertainties in terms of structure and parameters. These uncertainties cannot be described by deterministic models. The modelling and identification of nonlinear dynamic systems through the measured experimental data is a problem in engineering and technical processes. Therefore, field of system identification have become an important area of research. Fuzzy technology is an effective tool for dealing with complex nonlinear processes that are characterized with uncertain factors. In this paper, a novel approach based on Local Linear method learning in dynamical filter weights neurons for the identification of non-linear dynamic systems is presented. The fuzzy wavelet neural network combines wavelet theory with fuzzy logic and neural networks. Learning fuzzy rules and parameter update in fuzzy wavelet neural network is based on gradient decent method. The proposed approach is said to be Fuzzy Local Linear Wavelet Neural Network based model. It has been explained through examples. The structure is tested for the identification with both wavelet neural network and Fuzzy Local Linear Wavelet Neural Network that shows the comparative performance.KeywordsSystem IdentificationNon-linear SystemWavelet Neural NetworkFuzzy Wavelet Neural NetworkLocal Linear Wavelet Neural NetworkFuzzy Local Linear Wavelet Neural Network

An identification method of nonlinear dynamic system is studied in this paper. First, suppose that the original nonlinear dynamic system is described by Hammerstein model. Then, an intermediate model is generated. Next, Parameters of the intermediate model are obtained using a Bacterial Chemotaxis Optimization (BCO) approach. Finally, through the relationships of the parameters of intermediate model and those of Hammerstein model, we derive the parameters of the system. Consequently, the original nonlinear dynamic system is identified. The feasibility and efficiency of the presented algorithm are demonstrated using numerical simulations.

Linear stabilization of the programmed motions of non-linear controlled dynamical systems under parametric perturbations

It is well-known that the controls of neural network require that all the inputs of their input layer are available. Under this circumstance, the architecture of the neural network is constrained. Hence, this kind of neurocontrol cannot apply to a wide class of nonlinear and unknown dynamical systems, or the control system that requires more sensors. Although the unknown system state and the unknown system dynamics can be achieved from an adaptive observer and a learning neural network, respectively, the simultaneous existence of the estimated state error and the modeling error makes the control system more likely to be unstable. In this paper, a novel neurocontroller based on the concept of sliding mode with estimated state is constructed to tackle a wide class of unknown and nonlinear dynamical systems. The focal stability of the overall system can be verified by the Lyapunov stability criteria. Finally, simulations are presented to verify the usefulness of the proposed method.

This research frame work investigates the application of a clustered based Neuro-fuzzy system to nonlinear dynamic system modeling from a set of input-output training patterns. It is concentrated on the modeling via Takagi-Sugeno (T-S) modeling technique and the employment of fuzzy clustering to generate suitable initial membership functions. Hence, such created initial memberships are then employed to construct suitable T-S sub-models. Furthermore, the T-S fuzzy models have been validated and checked through the use of some standard model validation techniques (like the correlation functions). Compared to other well-known approximation techniques such as artificial neural networks, fuzzy systems provide a more transparent representation of the system under study, which is mainly due to the possible linguistic interpretation in the form of rules. Such intelligent modeling scheme is very useful once making complicated systems linguistically transparent in terms of fuzzy if-then rules. The developed T-S Fuzzy modeling system has been then applied to model a nonlinear antenna dynamic system with two coupled inputs and outputs. Validation results have resulted in a very close antenna sub-models of the original nonlinear antenna system. The suggested technique is very useful for development transparent linear control systems even for highly nonlinear dynamic systems.

In reality, the behavior and nature of nonlinear dynamical systems are ubiquitous in many practical engineering problems. The mathematical models of such problems are often governed by a set of coupled second-order differential equations to form multi-degree-of-freedom (MDOF) nonlinear dynamical systems. It is extremely difficult to find the exact and analytical solutions in general. In this paper, the homotopy analysis method is presented to derive the analytical approximation solutions for MDOF dynamical systems. Four illustrative examples are used to show the validity and accuracy of the homotopy analysis and modified homotopy analysis methods in solving MDOF dynamical systems. Comparisons are conducted between the analytical approximation and exact solutions. The results demonstrate that the HAM is an effective and robust technique for linear and nonlinear MDOF dynamical systems. The proof of convergence theorems for the present method is elucidated as well.

A method for identifying nonlinear dynamic systems with noise is proposed by using probabilistic universal learning networks (PrULNs). PrULNs are extensions of universal learning networks (ULNs). ULNs form a superset of neural networks and were proposed to provide a universal framework for modeling and control of nonlinear large-scale complex systems. But the ULN does not provide any stochastic characteristics of the signals propagating through it. The PrULNs are equipped with machinery to calculate stochastic properties of signals and to train network parameters so that the signals behave with the pre-specified stochastic properties. On the other hand it is generally recognized that there exists an overfitting problem when identification of nonlinear dynamic systems with noise is done by neural networks. In this paper, it is shown from simulation results of identification of a nonlinear robot dynamics that PrULNs are useful for avoiding the overfitting.

Teaching Aids for Modeling and Control of Hybrid Systems (CAMCHS)