Dynamical Systems and Methods

The shooting method is applied to prove that a damped pendulum with oscillatory forcing makes chaotic motions for certain parameters. The method is more intuitive than the Poincaré map method and provides more information about when the chaos occurs.

The transmission of energy between subsystems coupled in hybrid system is very important for different applications. For first as an introduction, by using the author’s previously published references and that of her students, a short survey of an analytical study of the energy transfer between coupled subsystems is presented as a basis of this chapter. An analytical study of the mechanical energy transfer between two coupled subsystems, as well as, between two or more coupled rotation motions is presented. For starting, an analytical analysis of the mechanical energy transfer between a linear and a nonlinear oscillators of a hybrid system (see Refs. by Hedrih (Stevanović) 2002 Hedrih [2006, 2005, 2006,?, 2003], Katica [2005], Hedrih [1997, 2005]) in the free, as well as forced, vibrations of a different types of interconnections between subsystems is presented. Coupling element between subsystems of the considered hybrid systems are standard light elements with elastic, viscoelastic, hereditary, or creeping properties as well as dynamical constrain element realized by rolling element with inertia properties. Using Krilov–Bogolyubov–Mitropolskiy’s asymptotic method, both the solutions in the first approximation and the system of nonlinear-coupled differential equations for the corresponding number of excited amplitudes and phases of multifrequency free as well as forced regimes are derived. By means of this asymptotic approximation of differential equations for the amplitudes and phases for forced vibrations of the coupled oscillators, the mutual influence of the nonlinear harmonics and energy transient were analyzed. The Lyapunov exponents corresponding to the each of two eigen like nonlinear modes are expressed by using energy of the corresponding eigen time components. A generalization of an analytical analysis of the transfer energy between linear and nonlinear oscillators for forced vibrations with different type constraints as a couple between two subsystems, each of them with one degree of freedom, is done. A mathematical analogy between discrete and complex discrete-continual hybrid systems is pointed out.In the second part, an analytical analysis is extended to the transfer energy between plates for free and forced transversal vibrations of a visco and nonlinear elastically connected double plate system. The analysis showed that the visco- and nonlinear elastic connection between plates caused the appearance of two-frequency like regime of time function, which corresponds to one eigen amplitude function of one mode, and also that time functions of different vibration modes are coupled, as well as energy transfer between plates in one eigen mode appear.Next, as an author’s new research result, an analytical study of the energy transfer between two coupled-like string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system in the free vibrations is presented.

This chapter is on the updated methodologies of diagnosing characteristics of nonlinear dynamic systems. The widely used and most applicable methods and approaches in characterizing the nonlinear behaviors of the systems are reviewed briefly. Characteristics of Lyapunov exponents and recently developed periodicity ratio approach are described in detail and compared. The applicability and efficiency of the approaches are presented and compared.

In this chapter, synchronization between two coupled populations of nonidentical phase oscillators is investigated. Generalizing the linear reformulation (Roberts Phys Rev E 77:031114, 2008), we could find explicit expressions of the synchronization order parameters, which include the phase coherence within each population and the phase coherence between two populations. Finally, numerical example is given to illustrate the theoretical results.

The chaotic synchronization of the Duffing oscillator and controlled pendulum is investigated. The analytical conditions for the partial and full synchronizations of the controlled pendulum with chaotic motions in the Duffing oscillator are discussed. The partial and full synchronizations are illustrated to show the analytical conditions. This synchronization is different from the controlled Duffing oscillator synchronizing with chaotic motion in the periodically excited pendulum.

The study of N-body systems and their simulation with various models always excited the scientific interest. Here we present an N-body model that has been under investigation in the last 10 years and is called the ring problem of (N+1) bodies, or otherwise the regular polygon problem of (N+1) bodies. In what follows, we give an overview of the scientific work that has been done through all these years, as well as the major results obtained so far.

Algorithms for the symbolic computation of polynomial conservation laws, generalized symmetries, and recursion operators for systems of nonlinear differential–difference equations (DDEs) are presented. The algorithms can be used to test the complete integrability of nonlinear DDEs. The ubiquitous Toda lattice illustrates the steps of the algorithms, which have been implemented in Mathematica. The codes InvariantsSymmetries.m and DDERecursionOperator.m can aid researchers interested in properties of nonlinear DDEs.

This chapter proposes an approximate polynomial solution for a nonlinear problem of the type: x ^′′ = F(x ^′ , x, t) on the [a,b] interval with initial conditions of the type: x ^′ (a) = x ₁, x(a) = x ₀, where F is a continuously differentiable real function. The approximate solution is expressed in terms of Taylor polynomials, whose coefficients are determined by solving a nonlinear system associated to the problem. The performance of the method is illustrated by two numerical examples.

The manuscript considers the dynamical symmetry usage to the integration of ODE. Symmetries with invariants guaranteeing the lowering of the ODE order are suggested. These symmetries include the whole class of point symmetries. The procedure for the dynamical symmetries finding is demonstrated. Concrete examples of the using of dynamical symmetries are given. The manuscript considers the application of the obtained solutions to the investigation of the nonlinear heat conduction equation.

The problems tackled in this chapter are directly or indirectly concerned with new trends in multitime optimal control. More precisley, we analyze the Riemannian convexity of energy functionals connected to the volumetric energy and the kinetic energy. The tools we use are those, not only of Riemannian geometry, but also of variational calculus and geometric dynamics. Some of the Lagrangians we discuss about are obtained by considering Euler–Lagrange prolongations of a PDE system of order one, such that, the solutions of the prolongation are ultra-potential maps. The problems solved here include: Riemannian convexity of kinetic energy functional, convex functions generated by convex functionals, invexity of kinetic and volumetric energy functionals, invexity for least square Lagrangians etc. A major result of this theory consists in relating the convexity of the energy functionals with the geometry of the underlying manifolds. The main key for our original approach is the new notion of sub-Killing vector fields on Riemannian manifolds.

We show that analogues of classical concepts of the Weyl limit point and limit circle cases can be introduced and investigated for second order linear dynamic equations on time scales. Since dynamical equations on time scales unifies and extends continuous and discrete dynamical equations (i.e., differential and difference equations), in this way, we establish more general theory of the limit point and limit circle cases.

In this manuscript, first we introduce some new condition, inspirit of Suzuki’s (C)-condition, on a self-mapping T on a subset K of a Banach space E. Secondly, we obtain some new fixed point theorems under these conditions.

We study a 4 ×4 complex matrix Jacobi (tri-diagonal matrix) arised from a non-Hermitian discrete quantum system. Reality of the eigenvalues of the matrix in question is investigated.

This paper is a partial summary of two plenary presentations by the author at two related conferences “Nonlinear Science and Complexity” and “New Trends in Nanotechnology and Nonlinear Dynamical Systems,” held in July 2010 in Ankara, Turkey. It outlines both well established and some recent achievements in asymptotic perturbation theory of solitary waves (solitons) and its applications to internal gravity waves in the ocean.

We analyze some properties of complex holomorphic flows and their related flows, obtaining original results. The problems solved include new representations of holomorphic flows, the hyperbolicity and stability of equilibrium points, the behavior near zeros and poles, potential or hamiltonian systems, etc. The tools are not only those of dynamical systems theory and complex functions, but also of differential geometry. The problems tackled are directly or indirectly concerned with new trends in mathematical literature dedicated to dynamics of complex holomorphic functions. One key to the new research results has been the interest for properties of special functions and their evolutions which appear in Applied Sciences.

In this paper a study is carried out to analyze the unsteady heat transfer effects due to a sudden introduction of heat source/sink on a steady viscous boundary layer MHD flow and heat transfer over a linearly stretching sheet subjected to a constant temperature. Governing boundary layer equations have been solved by an implicit finite difference method. Numerical results show that the steady state is reached quickly for a heat sink or for a large Prandtl number. The time to reach steady state increases under magnetic field. Upto a critical value of the strength of heat source, steady solution exists.

Pores and fractures in rocks are continuously being reshaped through different chemical and physical processes. Fluids filling the pore space and carrying different chemical species are responsible for these changes. In the present work, we study the effort of chemical kinetics on the reshaping of pore structure and thereby on permeability. A simulation study is carried on a two-dimentional random porous structure. The particles permeate with a constant Peclet number, and their diffusion is represented through a random walk. Changing the probability of interaction varies the strength of the chemical reaction between the fluid and the rock. This study is done for different concentrations for the active material in the fluid. A scaling law is found to exist between the changes in permeability with reaction rate.

Exciton–phonon dynamics on a 1D lattice with long-range exciton–exciton interaction have been introduced and elaborated. Long-range interaction leads to a nonlocal integral term in the motion equation of the exciton subsystem if we go from discrete to continuous space. In some particular cases for power-law interaction, the integral term can be expressed through a fractional order spatial derivative. A system of two coupled equations has been obtained, one is a fractional differential equation for the exciton subsystem, the other is a standard differential equation for the phonon subsystem. These two equations present a new fundamental framework to study nonlinear dynamics with long-range interaction. New approaches to model the impact of long-range interaction on nonlinear dynamics are: fractional generalization of Zakharov system, Hilbert–Zakharov system, Hilbert–Ginzburg–Landau equation and nonlinear Hilbert–Schrödinger equation. Nonlinear fractional Schrödinger equation and fractional Ginzburg–Landau equation are also part of this framework.

Spectral data for complex Jacobi matrices are introduced and the time evolution of the spectral data for the Jacobi matrix associated with the solution of the finite complex Toda lattice is computed.

Dynamic modeling of parallel manipulators is difficult due to the existing multiple closed-loops and kinematic constraints. Generally speaking, all dynamic modeling approaches use classical mechanics principles, leading to equivalent dynamic equations. Nevertheless, these equations may present different levels of complexity and computational loads. This chapter presents the generalized momentum approach to derive the inverse dynamic model of a Hexa-type fully-parallel manipulator. This approach showed high computational efficiency, measured by the number of arithmetic operations involved in the computation of the manipulator’s inertia and Coriolis and centripetal terms matrices.