Chaotic and deterministic behaviour of nonlinear geophysical fluid dynamics 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.

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

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

: Research was conducted which exploited the use of modern theoretical methods in nonlinear dynamical systems, fluid dynamics, and nonlinear distributed parameter control, to understand and control the dynamics of the turbulence process in turbulent boundary layer flows.

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.

Rotorcraft and their associated flight dynamics are highly non-linear systems, especially near extremities of the flight envelope. A global picture of the flight dynamics can be obtained by incorporating in the equations of motion non-linearities in aerodynamics, control system features and inertial terms. This paper deals with the coupling of these phenomena with the non-linear dynamics of an underslung load system. Dynamical systems theory provides the basis for the continuation and bifurcation method of analysing non-linear systems of ordinary differential equations, and forms the basis of this study. The principal strength of the bifurcation-based approach is to identify and give insight into phenomena that globally affect behaviour of an aircraft model, as the full system of non-linear equations is used in analysis. Implemented alongside appropriate numerical simulation runs, the technique promotes an understanding of the underlying structure governing the behaviour of non-linear dynamical systems. Bifurcation and continuation analysis is carried out here on a 21st order non-linear flight dynamics model representing a utility helicopter. Also, the equations of motion for an underslung load system are formulated and then analyzed in isolation. Thereafter they are appended to the helicopter model and a non-linear analysis of the complete system is performed. The effect of the underslung load dynamics on the nominal helicopter system is revealed by the bifurcation-based studies. The influence of varying parameters of the underslung model on the underlying equilibria of the rotorcraft is also investigated. The results reveal changes in helicopter and underslung load stability once the two systems are coupled. The underslung system reduces the extent of the stable equilibrium solution branches, and causes the stability augmentation system to saturate closer to the straight and level trim initial condition. The parameter dependance of the combined dynamics are presented and highlight areas of instability, which may be missed using conventional methods, and show how the transition between stable and unstable solutions is exactly determined as an underslung parameter is varied. This characterisation of the coupled system in terms of stability and response illustrates the potential for bifurcation-based methods to contribute to such non-linear analysis and synthesis problems.

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.

Three instruments which have proven useful for studying nonlinear dynamical systems are demonstrated. The 3D rotator enables one to display in real time any two-dimensional projection of an attractor in R/sup 3/; it can also be used to take Poincare sections of attractors. The laser projector is a laser-based X-Y projection oscilloscope for demonstrating bifurcation phenomena. The nonlinear dynamics analyzer characterizes steady-state behaviors of nonlinear dynamical systems by estimating the power spectrum, dimension, and Lyapunov exponents of the corresponding attractors from their time series. >

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.

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.

Reservoir computing is an effective model for learning and predicting nonlinear and chaotic dynamical systems; however, there remains a challenge in achieving a more dependable evolution for such systems. Based on the foundation of Koopman operator theory, considering the effectiveness of the sparse identification of nonlinear dynamics algorithm to construct candidate nonlinear libraries in the application of nonlinear data, an alternative reservoir computing method is proposed, which creates the linear Hilbert space of the nonlinear system by including nonlinear terms in the optimization process of reservoir computing, allowing for the application of linear optimization. We introduce an implementation that incorporates a polynomial transformation of arbitrary order when fitting the readout matrix. Constructing polynomial libraries with reservoir-state vectors as elements enhances the nonlinear representation of reservoir states and more easily captures the complexity of nonlinear systems. The Lorenz-63 system, the Lorenz-96 system, and the Kuramoto-Sivashinsky equation are used to validate the effectiveness of constructing polynomial libraries for reservoir states in the field of state-evolution prediction of nonlinear and chaotic dynamical systems. This study not only promotes the theoretical study of reservoir computing, but also provides a theoretical and practical method for the prediction of nonlinear and chaotic dynamical system evolution.

Many real-world systems exhibit complex dynamic nonlinear characteristics that cannot be modeled by typical statistical and machine learning models. The human cerebellum is a vital part of the brain system that possesses the capability to accurately model highly nonlinear physical dynamics. We can exploit our increasing knowledge of the human cerebellum to construct an intelligent computational model to effectively handle the complexity of nonlinear dynamic systems in the real world. This paper presents a novel brain-inspired computational model of the human cerebellum named the kernel density-based CMAC with Takagi-Sugeno-Kang fuzzy inference model (KCMAC-TSK) for fast and accurate nonlinear system identification. The structure of the KCMAC-TSK model is inspired by the neurophysiological aspects of cerebellar learning and development process. By incorporating a fuzzy model in KCMAC-TSK using kernel density estimation, we enhance the modeling capability, accuracy, and interpretability of the system. We applied the proposed KCMAC-TSK model in a challenging highway traffic flow modeling and prediction problem. Experimental results showed that KCMAC-TSK outperformed current modeling techniques, demonstrating the learning accuracy and effectiveness of KCMAC-TSK in handling complex nonlinear dynamic real-world systems.

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.