Homotopy Analysis Method for Multi-Degree-of-Freedom Nonlinear Dynamical Systems

Granular materials are extensively used throughout many industries. In countries like Australia and South Africa, national economies are critically dependent on the agricultural and mining industries, and it is important to understand bulk material behaviour in order to handle these materials properly. There are many diverse approaches to modelling granular materials. In this thesis, the Coulomb-Mohr continuum mechanical theory of granular materials is utilised, and some exact analytical solutions are presented for both stress and velocity fields. These solutions are formally valid provided sin φ = 1, where φ is the angle of internal friction, and they provide limiting solutions for so called highly frictional granular materials, for which sin φ ≈ 1. The solutions so obtained are exploited for three real industrial granular problems, namely gravity flow from converging hoppers, the stress distribution in stockpiles and finally, the stress distribution within rat-holes. The flow of granular materials, in the presence of gravity, through a converging hopper is an important problem that occurs in many industrial processes. Here, for highly frictional granular materials for both wedge and conical hopper flows, exact analytical solutions are obtained to determine the stress and velocity fields within the neighborhood of the hopper outlet. For the same problem, these exact solutions are exploited as the leading order terms in a regular perturbation series, where the correction terms are of order 1− sin φ. In this way, approximate analytical solutions are obtained that can be used to describe the behaviour of more realistic granular materials. The practice of storing granular materials in stockpiles occurs throughout the ii world in many industrial situations. In this thesis, exact solutions for highly frictional granular materials are used for the determination of the stress distributions within two-dimensional and axially symmetric stockpiles. Stockpile profiles considered here include the standard wedge and cone, as well as two-dimensional parabolic and axially symmetric cubic curved shaped stockpiles. We emphasise that curved stockpiles have not previously been considered in the literature. Our model assumes that the stockpile is composed of two regions, namely an inner rigid region and an outer yield region. The exact solution applies in the outer yield region, which is then extended continuously into the inner rigid region. A rat-hole is the general term used to describe those stable cavities, which frequently occur in storage hoppers and stockpiles, whose formation prevents further material falling through the outlet. In this thesis, we present exact solutions for both two-dimensional and axially symmetric rat-holes, where it is assumed that the highly frictional granular material has flowed through a central outlet, until the point is reached when there is no further flow of material. The remaining material is assumed to be in limiting equilibrium, that is static but on the point of yield. In both two dimensions and axial symmetry, two types of rat-holes are studied. The first type has a sloping upper surface of infinite extent, where the material is resting on a sloping rigid base containing an infinitesimal central outlet. The second type has a curved upper surface, where the material is resting between vertical walls and a horizontal rigid plane containing an infinitesimal central outlet. Again, we emphasise that neither of these rat-hole problems have previously been investigated in the literature.

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

Effects of fractional order time derivative on the solitary wave dynamics of the generalized ZK–Burgers equation

New and expressive analytical approximate solutions to resonance response of harmonically forced strongly odd nonlinear oscillators are proposed. This method combines Newton’s iteration with the harmonic balance method. Unlike the classical harmonic balance method, accurate and explicit analytical approximate solutions are established because linearization of the governing nonlinear differential equation is conducted prior to harmonic balancing. The approach yields simple linear algebraic equations instead of nonlinear algebraic equations which have no analytical solution. With carefully constructed corrective measures, only one single iteration is required to obtain very accurate analytical approximate solutions to resonance response. It is found that since determination of stability of the initial approximate solution that resulted from the single-term harmonic balance can lead to erroneous conclusions, correction to the solution is necessary. Three examples are presented to illustrate the applicability and effectiveness of the proposed technique. Specially, for oscillations in high-energy orbits of the bistable Duffing oscillator, the proposed method can also give excellent analytical approximate solutions.

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

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.

There have been the approximate analytic solution \\cite{Silveria} and several\napproximate analytic forms \\cite{0508156,Carroll,0303112} of the growth factor\n$D_{g}$ for the general dark energy models with the constant values of its\nequation of state $\\omde$ after Heath found the exact integral form of the\nsolution of $D_{g}$ for the Universe including the cosmological constant or the\ncurvature term. Recently, we obtained the exact analytic solutions of the\ngrowth factor for both $\\omde = -1$ or $-\\fr{1}{3}$ \\cite{SK1} and the general\ndark energy models with the constant equation of state $\\omde$ \\cite{SK3}\nindependently. We compare the exact analytic solution of $D_{g}$ with the other\nwell known approximate solutions. We also prove that the analytic solutions for\n$\\omde = -1$ or $-\\fr{1}{3}$ in Ref. \\cite{SK1} are the specific solutions of\nthe exact solutions of the growth factor for general $\\omde$ models in Ref.\n\\cite{SK3} even though they look quite different. We scrutinize the issue of\nusing the well known parameterizations of the growth index and its parameter\ngiven in Ref. \\cite{WS} to obtain the growth factor for general dark energy\nmodels. We also investigate the possible extensions of the exact solution of\n$D_{g}$ to the time-varying $\\omde$ for the comparison with observations.\n

In this study, an exact analytical solution for the convective heat transfer equation from a semi-spherical fin was presented. To obtain a mathematical model, the system was assumed to be a lump in the vertical direction and the governing equation in the Cartesian coordinate was transferred to the Mathieu equation. The exact solution was compared with numerical results such as the finite difference method and midpoint method with Richardson extrapolation (Midrich). Not surprisingly, the exact solution prevailed over the numerical solutions in terms of accuracy and ease of use. Furthermore, the effect of Biot number on the heat transfer of the fin and the fine performance was investigated. The relative error of the results obtained from the analytical and numerical solutions at the base, center, and tip of the fin was 0, 7.72, and 40.25 percent, respectively. The results showed that the relative error between the analytical and numerical solutions depends on the Biot number and varies as a function of the fin length. The obtained analytical solution could be encouraging from different mathematical and industrial applications' points of view.

In this work, approximate analytic solutions for different types of KdV equations are obtained using the homotopy analysis method (HAM). The convergence control parameter h helps us to adjust the convergence region of the approximate analytic solutions. The solutions are obtained in the form of power series. The obtained solutions and the exact solutions are shown graphically, highlighting the effects of non-linearity. We have compared the approximate analytical results which are determined by HAM, with the exact solutions and shown graphically with their absolute errors. By choosing an appropriate value of the convergence control parameter, we can obtain the solution in few iterations. All the computations have been performed using the software package MATHEMATICA.

In this paper, the analytical methods for approximate solutions of periodic motions to chaos in nonlinear dynamical systems are reviewed. Briefly discussed are the traditional analytical methods including the Lagrange stand form, perturbation methods, and method of averaging. A brief literature survey of approximate methods in application is completed, and the weakness of current existing approximate methods is also discussed. Based on the generalized harmonic balance, the analytical solutions of periodic motions in nonlinear dynamical systems with/without time-delay are reviewed, and the analytical solutions for period-m motion to quasi-periodic motion are discussed. The analytical bifurcation trees of period-1 motion to chaos are presented as an application.

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.

These are notes of the lectures presented by the author at the 1991 International Workshop on Application of Statistical Methods in Theoretical Physics and Fluid Dynamics, Indian Statistical Institute, Calcutta. After a very short introduction to the terminology, mathematical description, and geometrical view on nonlinear dynamical systems, a series of examples are presented. They illustrate some specific properties of such systems. Some of the examples have been published in local journals and are little known.

Exact and approximate solutions for randomly excited MDOF non-linear systems

In this paper, the homotopy perturbation method (HPM) combined with Padé approximant is applied to generate the approximate analytical solution of Korteweg de-Vries (KdV) equation. The results show that the combined method gives highly accurate approximate analytical solution. It is also found that this method can achieve more suitable results in comparison to HPM alone. The approximate solution is also compared with the exact solution and it is found to be in good agreement.

In this study, a nonlinear fractional logistic equations is proposed in the context of a modified form of the conformable fractional-order derivative. The analytical and approximate-analytical solutions are obtained for the proposed mathematical model via the method of conformable fractional differential transform method (CFDTM) and all obtained results have been analyzed and compared with homotopy perturbation method (HPM). With plotting the figures, we compared the CFDTM, HPM, and exact solution together and conclude that the CFDTM solution is closer than the exact solution in comparison with the HPM. The methods are illustrated by two examples of FOLE and solution is obtained. All obtained solutions have been analyzed and compared graphically to validate the effiency and applicability of all results.