Solutions Of Nonlinear Dynamical Systems Research Articles

A high efficiency Lie derivative algorithm for the nonautonomous nonlinear systems

Numerical procedure plays a key role in tackling the solutions of nonlinear dynamical systems. With the advent of the age of big data and high-power computing, developing efficient and fast numerical algorithms is an urgent task. This paper extends the Lie derivative discretization algorithm to the nonautonomous nonlinear systems and investigates the numerical solutions of the systems. The periodic solutions of three different classical nonlinear systems are calculated, and the results are compared to those values calculated from the Runge–Kutta fourth-order algorithm, which demonstrated that the Lie derivative algorithm has the advantages of large time step and short computation time.

Decision-making on the solution of non-linear dynamical systems of Kannan non-expansive type in Nakano sequence space of fuzzy numbers

Decision-making on the solution of non-linear dynamical systems of Kannan non-expansive type in Nakano sequence space of fuzzy numbers

Are Chebyshev-based stability analysis and Urabe’s error bound useful features for Harmonic Balance?

Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors’ knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe’s error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.

Making Decisions about How to Solve Nonlinear Dynamical Systems of the Kannan Type in a New Complex Function Space of Formal Power Series

The sufficient settings of the space generated by absolute type-weighted gamma matrices of rank p in the Nakano complex functions of the formal power series, as well as its associated prequasioperators’ ideal equipped with definite functions, are defined and explained in this paper. Assorted prequasinorms are shown to have the Fatou characteristic. Some of its geometric and topological features and the prequasioperators’ ideal that goes with it are talked about. These structures have a relationship with the fixed points of the Kannan contraction and nonexpansive mappings. By looking at real-world examples and how they are used, it is shown that there are solutions for nonlinear dynamical systems of the Kannan type.

Collocation‐based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

AbstractPeriodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely‐used approaches for such systems, but are either confined to low‐order approximations or impaired by aliasing and improper‐sampling problems. Here we propose a collocation‐based harmonic balance framework to successfully unify and reconstruct the HB‐like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency‐domain and time‐domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high‐order (>100) nonaliasing solutions, previously deemed out‐of‐reach, for a range of complex nonlinear systems including the cavitation bubble dynamics, the three‐body problem and the two dimensional airfoil dynamics. We show that the present method is 2–3 orders of magnitude more accurate and simultaneously much faster than the state‐of‐the‐art method. Hence, it has immediate applications in multidisciplinary problems where highly accurate periodic solutions are sought.

Mathematical study of neural feedback roles in small target motion detection.

Building an efficient and reliable small target motion detection visual system is challenging for artificial intelligence robotics because a small target only occupies few pixels and hardly displays visual features in images. Biological visual systems that have evolved over millions of years could be ideal templates for designing artificial visual systems. Insects benefit from a class of specialized neurons, called small target motion detectors (STMDs), which endow them with an excellent ability to detect small moving targets against a cluttered dynamic environment. Some bio-inspired models featured in feed-forward information processing architectures have been proposed to imitate the functions of the STMD neurons. However, feedback, a crucial mechanism for visual system regulation, has not been investigated deeply in the STMD-based neural circuits and its roles in small target motion detection remain unclear. In this paper, we propose a time-delay feedback STMD model for small target motion detection in complex backgrounds. The main contributions of this study are as follows. First, a feedback pathway is designed by transmitting information from output-layer neurons to lower-layer interneurons in the STMD pathway and the role of the feedback is analyzed from the view of mathematical analysis. Second, to estimate the feedback constant, the existence and uniqueness of solutions for nonlinear dynamical systems formed by feedback loop are analyzed via Schauder's fixed point theorem and contraction mapping theorem. Finally, an iterative algorithm is designed to solve the nonlinear problem and the performance of the proposed model is tested by experiments. Experimental results demonstrate that the feedback is able to weaken background false positives while maintaining a minor effect on small targets. It outperforms existing STMD-based models regarding the accuracy of fast-moving small target detection in visual clutter. The proposed feedback approach could inspire the relevant modeling of robust motion perception robotics visual systems.

Analytic Matrix Method for Frequency Response Techniques Applied to Nonlinear Dynamical Systems II: Large Amplitude Oscillations

This work is the second in a series of articles that deal with analytical solutions of nonlinear dynamical systems under oscillatory input that may exhibit harmonic frequencies. Frequency response techniques of nonlinear dynamical systems are usually analyzed with numerical methods, because in most cases analytical solutions such as the harmonic balance series solution turn out to be difficult, if not impossible, as they are based on an infinite series of trigonometric functions with harmonic frequencies. The key contribution of the analytic matrix methods reported in the present series of articles is to work with the invariant submanifold of the problem and to propose the solution as infinite power series of the oscillatory input; this procedure is a direct one that speeds up the computations compared to traditional series solution methods. The method reported in the first contribution of this series allows for the computation of the analytical solution only for small and medium amplitudes of the oscillatory input, because these series may diverge when large amplitudes are applied. Therefore, the analytic matrix method reported here, which is a reconfiguration of the method proposed in the first contribution in this series, allows the solving of problems in the regime of large-amplitude oscillations where the contributions of the high order harmonics affect the amplitudes of the low order harmonics, leading to amplitude- and frequency-dependent coefficients for the infinite series of trigonometric function expansion.

A novel criterion for global incremental stability of dynamical systems

In this paper, we establish the sufficient conditions guaranteeing global uniform exponential stability, or at least global asymptotic stability, of all solutions for nonlinear dynamical systems, also known as global incremental stability (GIS) of the systems. We provide here an alternative approach for assessment of GIS in terms of logarithmic norm under which the stability becomes a topological notion and also generalize both horizontally and vertically the well-known Demidovich criterion for GIS of dynamical systems. Convergence of all solutions to the origin x=0, which is not assumed to be an equilibrium state of system, is also analyzed. Theory is illustrated by a simulation experiment.

Численно-аналитический метод преобразований для анализа нелинейных математических моделей полиномиальной структуры

The article considers a scientific problem of investigating mathematical models with polynomial nonlinearity, which are represented by systems of nonlinear differential equations. There is presented a numericall-analytical transformation method for investigating nonlinear dynamical systems with many degrees of freedom of polynomial structure. As opposed to the analogues, this method allows solving a wide range of problems for nonlinear systems 
 of general polynomial structure while reducing the computational resource intensity. An algorithm of the method 
 of transformations for investigating the nonlinear dynamic systems with m degrees of freedom is given. The research of stability of solutions of nonlinear dynamic systems with m degrees of freedom is carried out. Algorithmic formulas of transformation method for the solution of systems with nonlinearity of the fourth degree are given. Computational experiments on solving systems of differential equations with a small nonlinear part in the form of a multinomial 
 of the sixth degree using the transformation method show fourth order accuracy in computation. A nonlinear mathematical model of the vibration protection system is investigated by the presented method of transformations. The theorem on determination of the stationary state for systems of differential equations of polynomial structure by transformation method is proved. Algorithmic formulas for computation are presented. A general matrix form for vector indices is presented. Formulas for economical computation of right-hand sides of polynomial structure are presented and it is proposed to apply Pan's scheme with coefficient preprocessing. This method allows studying the different modes 
 of nonlinear models dynamics, for example, to determine such extreme modes as resonance, sub-harmonic, and polyharmonic modes. As an example of the method application, the vibration protection problem of the tower from external periodic influences was solved. The transformed solution takes into account all nonlinear polynomial components. The method allows to investigate the dynamics of a wide range of nonlinear systems with the necessary accuracy

A convolutional-iterative solver for nonlinear dynamical systems

In this paper, a new solver is developed to obtain the solution of nonlinear dynamical systems. The method is based on alternative field equations in which the convolution product appears in place of dot products. The technique is higher-order accurate in the sense that the accuracy is not lost by increasing the time-step, conserving constants of motion. Several examples, including bilinear hardening and softening mass–spring systems and a nonlinear elastic pendulum, are considered to show the validity of the new method.

Nonlinear analysis of two-degree of freedom system with nonlinear springs

In this paper, an analytic solving procedure for a two degree of freedom oscillatory system (2DOF) is considered. Vibration of the two-mass system is mathematically modeled with two coupled second-order equations. The system is assumed to be strongly nonlinear, and a high order Homotopy Perturbation Method (HPM) for solving the system is developed. The frequency of vibration is obtained in the second approximation. To validate the method, two examples are considered. The solutions of the HPM are compared with those of Newton–harmonic balance (NHB) and numerical approaches. It has been demonstrated that HPM leads to a highly accurate solution even with the first iteration of the solution and is valid for a wide range of vibration amplitudes, showing promise as a means to provide an exact or closed-form solution of nonlinear dynamical systems, as indicated in the following examples.

Undefined Frequencies Method

A method of analytical construction of solutions of nonlinear dynamic systems is introduced for the case where solutions have oscillatory type. The method does not require any a priori assumptions on the frequencies of oscillations that are defined on course of successive approximations. The method can be applied without assuming the existence of some small parameter, and this is illustrated by the analysis of particular dynamic systems.

Control and numerical analysis for cancer chaotic system

This article investigates the problem of control of chaotic dynamics of tumor cells, immune system cells, and healthy tissue cells in a three-dimensional cancer model by adaptive control technique. Adaptive control law is derived such that the trajectory of controlled system asymptotically approaches equilibrium point with estimated parameter converges to stabilizing values. A nonlinear control law is designed which change the original chaotic system into a controlled one linear system. In addition, we present and analyze numerical solution of the cancer dynamical system with the help of a discretization technique. Achieved solutions show a comparable results with Runge–Kutta methods. The reliability and accuracy of the proposed technique is presented by comparing numerical results. The used technique has displayed a brilliant prospective in dealing with the numerical solutions of nonlinear dynamical systems.

Avoidance of artifacts in harmonic balance solutions for nonlinear dynamical systems

Avoidance of artifacts in harmonic balance solutions for nonlinear dynamical systems

Eigenvalue Based Approach for Assessment of Global Robustness of Nonlinear Dynamical Systems

In this paper we have established the sufficient conditions for asymptotic convergence of all solutions of nonlinear dynamical system (with potentially unknown and unbounded external disturbances) to zero with time t → ∞ . We showed here that the symmetric part of linear part of nonlinear nominal system, or, to be more precise, its time-dependent eigenvalues, play important role in assessment of the robustness of systems.

A Fractional B-spline Collocation Method for the Numerical Solution of Fractional Predator-Prey Models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost.

Nonstationary solutions of nonlinear dynamical systems excited by Gaussian white noise

Nonlinear dynamical systems to random excitations may fail long before stationarity is achieved. Transient state has to be taken into account. A novel approximate technique for determining nonstationary probability density function (PDF) of randomly excited nonlinear Oscillator is developed. Specifically, it expresses the PDF approximation in terms of polynomial functions with time-dependent coefficients. By applying statistical linearization and weighted residual method, residual error of the FPK equation associated with approximated solution is reduced to a series of nonlinear first-order ordinary differential equations, which can be solved by the numerical method. Finally, a class of nonlinear vibrating systems with additive excitations or/and parametric excitations are considered. The obtained PDF has tail regions of logarithmic form, which are important for reliability and failure analysis, and agrees very well with the simulated ones. In particular, the computational time spent by the proposed procedure is a very small fraction of the one taken by the MCS method. This technique can be used as a convenient tool for assessing the accuracy of alternative, more general, approximate solution methods.

Improving Wilson-θ and Newmark-β Methods for Quasi-Periodic Solutions of Nonlinear Dynamical Systems

Quasi-periodic responses can appear in a wide variety of nonlinear dynamical systems. To the best of our knowledge, it has been a tough job for years to solve quasi-periodic solutions, even by numerical algorithms. Here in this paper, we will present effective and accurate algorithms for quasi-periodic solutions by improving Wilson-θ and Newmark-β methods, respectively. In both the two methods, routinely, the considered equations are rearranged in the form of incremental equilibrium equations with the coefficient matrixes being updated in each time step. In this study, the two methods are improved via a predictor-corrector algorithm without updating the coefficient matrixes, in which the predicted solution at one time point can be corrected to the true one at the next. Numerical examples show that, both the improved Wilson-θ and Newmark-β methods can provide much more accurate quasi-periodic solutions with a smaller amount of computational resources. With a simple way to adjust the convergence of the iterations, the improved methods can even solve some quasi-periodic systems effectively, for which the original methods cease to be valid.

A variational numerical method based on finite elements for the nonlinear solution characteristics of the periodically forced Chen system

Nonlinear dynamical systems and their solutions are very sensitive to initial conditions and therefore need to be approximated carefully. In this article, we present and analyze nonlinear solution characteristics of the periodically forced Chen system with the application of a variational method based on the concept of finite time-elements. Our approach is based on the discretization of physical time space into finite elements where each time-element is mapped to a natural time space. The solution of the system is then determined in natural time space using a set of suitable basis functions. The numerical algorithm is presented and implemented to compute and analyze nonlinear behavior at different time-step sizes. The obtained results show an excellent agreement with the classical RK-4 and RK-5 methods. The accuracy and convergence of the method is shown by comparing numerically computed results with the exact solution for a test problem. The presented method has shown a great potential in dealing with the solutions of nonlinear dynamical systems and thus can be utilized in delineating different features and characteristics of their solutions.

A Modified Runge–Kutta Method for Nonlinear Dynamical Systems With Conserved Quantities

In this paper, explicit Runge–Kutta methods are investigated for numerical solutions of nonlinear dynamical systems with conserved quantities. The concept, ε-preserving is introduced to describe the conserved quantities being approximately retained. Then, a modified version of explicit Runge–Kutta methods based on the optimization technique is presented. With respect to the computational effort, the modified Runge–Kutta method is superior to implicit numerical methods in the literature. The order of the modified Runge–Kutta method is the same as the standard Runge–Kutta method, but it is superior in preserving the conserved quantities to the standard one. Numerical experiments are provided to illustrate the effectiveness of the modified Runge–Kutta method.