Thermomechanical Dynamics (TMD) and Bifurcation-Integration Solutions in Nonlinear Differential Equations with Time-Dependent Coefficients

The new independent solutions of the nonlinear differential equation with time-dependent coefficients (NDE-TC) are discussed, for the first time, by employing experimental device called a drinking bird whose simple back-and-forth motion develops into water drinking motion. The solution to a drinking bird equation of motion manifests itself the transition from thermodynamic equilibrium to nonequilibrium irreversible states. The independent solution signifying a nonequilibrium thermal state seems to be constructed as if two independent bifurcation solutions are synthesized, and so, the solution is tentatively termed as the bifurcation-integration solution. The bifurcation-integration solution expresses the transition from mechanical and thermodynamic equilibrium to a nonequilibrium irreversible state, which is explicitly shown by the nonlinear differential equation with time-dependent coefficients (NDE-TC). The analysis established a new theoretical approach to nonequilibrium irreversible states, thermomechanical dynamics (TMD). The TMD method enables one to obtain thermodynamically consistent and time-dependent progresses of thermodynamic quantities, by employing the bifurcation-integration solutions of NDE-TC. We hope that the basic properties of bifurcation-integration solutions will be studied and investigated further in mathematics, physics, chemistry and nonlinear sciences in general.

The Solution to Impulse Boundary Value Problem for a Class of Nonlinear Fractional Functional Differential Equations

In this paper, we investigate the existence of solution for a class of impulse boundary value problem of nonlinear fractional functional differential equation of mixed type. We obtain the existence results of solution by applying some well-known fixed point theorems. An example is given to illustrate the effectiveness of our result.

OSCILLATORY BEHAVIOR FOR HIGH ORDER FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, the oscillatory behavior for high order nonlinear functional differential equations are studied by means of the Lebesgue measure. It is found that the nonoscillatory solutions only have two kinds on some conditions. And necessary conditions for the existence of each kind of nonoscillatory solutions are presented as well. At the same ime, some sufficient conditions for oscillatory solutions are also established.

DYNAMIC BIFURCATION OF NONLINEAR EVOLUTION EQUATIONS

The authors introduce a notion of dynamic bifurcation for nonlinear evolution equa- tions, which can be called attractor bifurcation. It is proved that as the control pa- rameter crosses certain critical value, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, where m + 1 is the number of eigenvalues crossing the imaginary axis. The attractor bifurcation theory presented in this article generalizes the existing steady state bifurcations and the Hopf bifurcations. It provides a uni?ed point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics.

THE EXACT MEROMORPHIC SOLUTIONS OF SOME NONLINEAR DIFFERENTIAL EQUATIONS

We find the exact forms of meromorphic solutions of the nonlinear differential equations■,n≥3,k≥1,where q,Q are nonzero polynomials,Q■Const.,and p_(1),p_(2),α_(1),α_(2)are nonzero constants withα_(1)≠α_(2).Compared with previous results on the equation p(z)f^(3)+q(z)f"=-sinα(z)with polynomial coefficients,our results show that the coefficient of the term f^((k))perturbed by multiplying an exponential function will affect the structure of its solutions.

Solitary,periodic,kink wave solutions of a perturbed high-order nonlinear Schrödinger equation via bifurcation theory

In this paper,by using the bifurcation theory for dynamical system,we construct traveling wave solutions of a high-order nonlinear Schrödinger equation with a quintic nonlin-earity.Firstly,based on wave variables,the equation is transformed into an ordinary differential equation.Then,under the parameter conditions,we obtain the Hamiltonian system and phase portraits.Finally,traveling wave solutions which contains solitary,periodic and kink wave so-lutions are constructed by integrating along the homoclinic or heteroclinic orbits.In addition,by choosing appropriate values to parameters,different types of structures of solutions can be displayed graphically.Moreover,the computational work and it’sfigures show that this tech-nique is influential and efficient.

A Comparative Study of Adomian Decomposition Method with Variational Iteration Method for Solving Linear and Nonlinear Differential Equations

This study compares the Adomian Decomposition Method (ADM) and the Variational Iteration Method (VIM) for solving nonlinear differential equations in engineering. Differential equations are essential for modeling dynamic systems in various disciplines, including biological processes, heat transfer, and control systems. This study addresses first, second, and third-order nonlinear differential equations using Mathematica for data generation and graphing. The ADM, developed by George Adomian, uses Adomian polynomials to handle nonlinear terms, which can be computationally intensive. In contrast, VIM, developed by He, directly iterates the correction functional, providing a more straightforward and efficient approach. This study highlights VIM’s rapid convergence and effectiveness of VIM, particularly for nonlinear problems, where it simplifies calculations and offers direct solutions without polynomial derivation. The results demonstrate VIM’s superior efficiency and rapid convergence of VIM compared with ADM. The VIM’s minimal computational requirements make it practical for real-time applications and complex system modeling. Our findings align with those of previous research, confirming VIM’s efficiency of VIM in various engineering applications. This study emphasizes the importance of selecting appropriate methods based on specific problem requirements. While ADM is valuable for certain nonlinearities, VIM’s approach is ideal for many engineering scenarios. Future research should explore broader applications and hybrid methods to enhance the solution’s accuracy and efficiency. This comprehensive comparison provides valuable guidance for selecting effective numerical methods for differential equations in engineering.

Hopf Bifurcation of a Gene-Protein Network Module with Reaction Diffusion and Delay Effects

The infinite dimensional partial delay differential equation is set forth and delay difference state feedback control is considered to describe the cell cycle growth in eukaryotic cell cycles. Hopf bifurcation occurs as varying free parameters and time delay continuously and the multi-layer oscillation phenomena of the homogeneous steady state of a simple gene-protein network module is investigated. Normal form is derived based on normal formal analysis technique combined with center manifold theory, which is further to compute the bifurcating direction and the stability of bifurcation periodical solutions underlying Hopf bifurcation. Finally, the numerical simulation oscillation phenomena is in coincidence with the theoretical analysis results.

Contractivity and Exponential Stability of Solutions to Nonlinear Neutral Functional Differential Equations in Banach Spaces

A series of eontractivity and exponential stability results for the solutions to nonlinear neutral functional differential equations （NFDEs） in Banach spaces are obtained, which provide unified theoretical foundation for the contractivity analysis of solutions to nonlinear problems in functional differential equations （FDEs）, neutral delay differential equations （NDDEs） and NFDEs of other types which appear in practice.

SOME OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR FUNCTIONAL ORDINARY DIFFERENTIAL EQUATIONS

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations（a（t）x＇（t））＇＋δ1p（t）x＇（t） ＋δ2q（t）f（x（g（t））） = 0,for 0 ≤ to≤ t, where 51 = ：El and δ±1. The functions p,q,g ： [t0, ∞） → R, f ： R → are continuous, a（t） 〉 0,p（t） ≥0,q（t） 〉 0 for t ≥ to,lirn g（t） = ∞, and q is not identically zero on any subinterval of [to, ∞）. Moreover, the functions q（t）, g（t）, and a（t） are continuously differentiable.

EXISTENCE AND UNIQUENESS OF THE SOLUTION OF A CLASS OF NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

The unicity of the solution, if any, of a class of nonlinear functional differential equations (fde) is established with the help of a transformation. The transformation reduces the fde to an ordinary differential equation. Existence of the solution is established by means of a fixed point theorem.

Stability analysis of solutions to nonlinear stiff Volterra functional differential equations in Banach spaces

A series of stability, contractivity and asymptotic stability results of the solutions to nonlinear stiff Volterra functional differential equations (VFDEs) in Banach spaces is obtained, which provides the unified theoretical foundation for the stability analysis of solutions to nonlinear stiff problems in ordinary differential equations(ODEs), delay differential equations(DDEs), integro-differential equations(IDEs) and VFDEs of other type which appear in practice.

Nonlinear Differential Equation of Macroeconomic Dynamics for Long-Term Forecasting of Economic Development

In this article we derive a general differential equation that describes long-term economic growth in terms of cyclical and trend components. Equation is based on the model of non-linear accelerator of induced investment. A scheme is proposed for obtaining approximate solutions of nonlinear differential equation by splitting solution into the rapidly oscillating business cycles and slowly varying trend using Krylov-Bogoliubov-Mitropolsky averaging. Simplest modes of the economic system are described. Characteristics of the bifurcation point are found and bifurcation phenomenon is interpreted as loss of stability making the economic system available to structural change and accepting innovations. System being in a nonequilibrium state has a dynamics with self-sustained undamped oscillations. The model is verified with economic development of the US during the fifth Kondratieff cycle (1982-2010). Model adequately describes real process of economic growth in both quantitative and qualitative aspects. It is one of major results that the model gives a rough estimation of critical points of system stability loss and falling into a crisis recession. The model is used to forecast the macroeconomic dynamics of the US during the sixth Kondratieff cycle (2018-2050). For this forecast we use fixed production capital functional dependence on a long-term Kondratieff cycle and medium-term Juglar and Kuznets cycles. More accurate estimations of the time of crisis and recession are based on the model of accelerating log-periodic oscillations. The explosive growth of the prices of highly liquid commodities such as gold and oil is taken as real predictors of the global financial crisis. The second wave of crisis is expected to come in June 2011.

Bellman Equation for Optimal Processes with Nonlinear Multi-Parametric Binary Dynamic System

A process represented by nonlinear multi-parametric binary dynamic system is investigated in this work. This process is characterized by the pseudo Boolean objective functional. Since the transfer functions on the process are Boolean functions, the optimal control problem related to the process can be solved by relating between the transfer functions and the objective functional. An analogue of Bellman function for the optimal control problem mentioned is defined and consequently suitable Bellman equation is constructed.

On Universality of Transition to Chaos Scenario in Nonlinear Systems of Ordinary Differential Equations of Shilnikov’s Type

Several nonlinear three-dimensional systems of ordinary differential equations are studied analytically and numerically in this paper in accordance with universal bifurcation theory of Feigenbaum-Sharkovskii-Magnitsky [1] [2]. All systems are autonomous and dissipative and display chaotic behaviour. The analysis confirms that transition to chaos in such systems is performed through cascades of bifurcations of regular attractors.

NONLINEAR DYNAMICS OF AXIALLY ACCELERATING VISCOELASTIC BEAMS BASED ON DIFFERENTIAL QUADRATURE

This paper investigates nonlinear dynamical behaviors in transverse motion of an axially accelerating viscoelastic beam via the differential quadrature method. The governing equation, a nonlinear partial-differential equation, is derived from the viscoelastic constitution relation using the material derivative. The differential quadrature scheme is developed to solve numerically the governing equation. Based on the numerical solutions, the nonlinear dynamical behaviors are identified by use of the Poincare map and the phase portrait. The bifurcation diagrams are presented in the case that the mean axial speed and the amplitude of the speed fluctuation are respectively varied while other parameters are fixed. The Lyapunov exponent and the initial value sensitivity of the different points of the beam, calculated from the time series based on the numerical solutions, are used to indicate periodic motions or chaotic motions occurring in the transverse motion of the axially accelerating viscoelastic beam.

Necessary and sufficient conditions for the existence of equilibrium in abstract non-autonomous functional differential equations

non-autonomous finite-delay functional differential equations without any monotone conditions assumed.A minimal set is constructed in terms of which necessary and sufficient conditions for a continuous equilibrium to exist are also obtained.Several illustrative examples are employed to demonstrate our results.

Existence of Positive Solutions of Three-point Boundary Value Problem for Higher Order Nonlinear Fractional Differential Equations

In this paper, we consider the positive solutions of fractional three-point boundary value problem of the form Dο^α＋u（t）＋f（t,u（t）,u＇（t）,…,u^（n-3）（5）,u^（n-2）（t））=0,u^（i）（0）=0,0≤i≤n-2,u^（n-2）（1）-βu^（n-2）（ξ）=0,where 0〈t〈1,n-1〈α≤n,n≥2,ξ Е（0,1）,βξ^a-n〈1. We first transform it into another equivalent boundary value problem. Then, we derive the Green＇s function for the equivalent boundary value problem and show that it satisfies certain properties. At last, by using some fixed-point theorems, we obtain the existence of positive solution for this problem. Example is given to illustrate the effectiveness of our result.

New lump solutions and several interaction solutions and their dynamics of a generalized(3+1)-dimensional nonlinear differential equation

In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

Dynamics of traveling wave solutions to a highly nonlinear Fujimoto–Watanabe equation

In this work, we apply the bifurcation method of dynamical systems to investigate the underlying complex dynamics of traveling wave solutions to a highly nonlinear Fujimoto–Watanabe equation. We identify all bifurcation conditions and phase portraits of the system in different regions of the three-dimensional parametric space, from which we present the sufficient conditions to guarantee the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, kink-like(antikink-like) wave solutions, and compactons. Furthermore, we obtain their exact expressions and simulations, which can help us understand the underlying physical behaviors of traveling wave solutions to the equation.