Universal theory of dynamical chaos in nonlinear dissipative systems of differential equations

A new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations including ordinary and partial, autonomous and non-autonomous differential equations and differential equations with delay arguments is presented in this paper. Four corner-stones lie in the foundation of this theory: the Feigenbaum’s theory of period doubling bifurcations in one-dimensional mappings, the Sharkovskii’s theory of bifurcations of cycles of an arbitrary period up to the cycle of period three in one-dimensional mappings, the Magnitskii’s theory of rotor type singular points of two-dimensional non-autonomous systems of differential equations as a bridge between one-dimensional mappings and differential equations and the theory of homoclinic cascade of bifurcations of stable cycles in nonlinear differential equations. All propositions of the theory are strictly proved and illustrated by numerous analytical and computing examples.     

Chaotic attractors in nonlinear dissipative systems of ordinary differential equations

A mechanism is proposed describing the formation of irregular attractors in a wide class of three-dimensional nonlinear autonomous dissipative systems of ordinary differential equations with singular cycles. The attractors of such systems, called singular attractors, lie on two-dimensional surfaces in the phase space and have no positive Lyapunov exponents. In all systems of this class the onset of chaos follows the same universal mechanism: a cascade of Feigenbaum’s period doubling bifurcations, a subharmonic cascade of Sharkovskii’s bifurcations, and eventually a homoclinic cascade. All classical chaotic systems, including Lorenz, Rössler, and Chua systems, satisfy these conditions.     

On the topological structure of singular attractors of nonlinear systems of differential equations

We study the topological structure of singular (in the sense of the Feigenbaum-Sharkovskii-Magnitskii theory) attractors of nonlinear dissipative systems of differential equations. We show that any such attractor is a stable nonperiodic trajectory lying on a two-dimensional infinitely folded heteroclinic separatrix manifold generated by the unstable two-dimensional invariant manifold of the original singular cycle as the bifurcation parameter of the system varies. The results obtained for two-dimensional nonautonomous and three-dimensional autonomous dissipative systems are generalized to autonomous multi- and infinite-dimensional dissipative systems as well as to conservative (in particular, Hamiltonian) systems.     

Investigation of saddle-node and pitchfork bifurcations by Magnitskii’s stabilization method

We consider Magnitskii’s method in almost original form and prove a property of it, on the basis of which we suggest a way to use the method for stabilizing saddle cycles and cycles passing through a pitchfork bifurcation. By way of example, we consider some solutions of the four-dimensional Yang-Mills equations in the presence of the Higgs mechanism.     

Positivity conditions and bounds for Green’s functions for higher order two-point BVP

We consider a linear nonautonomous higher order ordinary differential equation and establish the positivity conditions and two-sided bounds for Green’s function for the two-point boundary value problem. Applications of the obtained results to nonlinear equations are also discussed.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

Asymptotic representations of solutions of one class of nonlinear nonautonomous differential equations of the third order

We establish asymptotic representations for unbounded solutions of nonlinear nonautonomous differential equations of the third order that are close, in a certain sense, to equations of the Emden-Fowler type. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1363–1375, October, 2007.     

Some Remarks on Distributed PCHD-Systems

Port controlled Hamiltonian systems with dissipation are a well known tool for the modeling and the controller design for plants, described by nonlinear ordinary differential equations. This contribution presents a possible extension to systems, described by partial differential equations, where the state manifold and the input space of the ODE case are replaced by new geometric structures. This approach takes dissipative effects into account and shows, how distributed ports can be introduced. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite difference technique for solving optimization problems governed by linear functional differential equations

Aspects of the approximation and optimal control of systems governed by linear retarded nonautonomous functional differential equations (FDE) are considered. First, certain FDE are shown to be equivalent to corresponding abstract ordinary differential equations (ODE). Next, it is demonstrated that these abstract ODE may be approximated by difference equations in finite dimensional spaces. The optimal control problem for systems governed by FDE is then reduced to a sequence of mathematical programming problems. Finally, numerical results for two examples are presented and discussed.     

Asymptotic representations of the solutions of essentially nonlinear nonautonomous second-order differential equations

We establish asymptotic representations for the solutions of a class of nonlinear nonautonomous second-order differential equations. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 310–331, March, 2008.     

Transition to chaos in nonlinear dynamical systems described by ordinary differential equations

We investigate scenarios that create chaotic attractors in systems of ordinary differential equations (Vallis, Rikitaki, Rossler, etc.). We show that the creation of chaotic attractors is governed by the same mechanisms. The Feigenbaum bifurcation cascade is shown to be universal, while subharmonic and homoclinic cascades may be complete, incomplete, or not exist at all depending on system parameters. The existence of a saddle-focus equilibrium plays an important and possibly decisive role in the creation of chaotic attractors in dissipative nonlinear systems described by ordinary differential equations. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 3, pp. 73–98, 2003.     

Global dynamics of one class of nonlinear nonautonomous systems with time-varying delays

A class of nonautonomous systems of nonlinear delay differential equations was studied via construction of matrix inequalities and comparison techniques. The results for the nonautonomous systems with time-varying delays are novel, e.g., the global stability of differential equations with nonlinear (casual) Volterra operators is considered for the first time in the literature. Criteria obtained for permanence and global attractivity are explicit and hence are convenient for applying/verifying in practice. We illustrate applications of the results obtained to the nonautonomous and asymptotically autonomous Nicholson-type models.     

Remarks on some systems of nonlinear Schrödinger equations

We prove the existence of solutions of some nonautonomous systems of nonlinear Schr?dinger equations, by means of perturbation techniques. The work has been supported by M.U.R.S.T. under the national project “Variational methods and nonlinear differential equations”.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

Foreword

Steve Smale set the agenda for FoCM in his call for the 1995 conference in Park City, Utah. No stranger he to ambitious agendas and extraordinary accomplishments. He is one of the dominant figures in the mathematics of the second half of the twentieth century. Smale’s theory of immersions, the generalized Poincare conjecture, and H-cobordism theorems with their far-reaching consequences are the bedrock of differential topology. His horseshoe is the hallmark of chaos, and his hyperbolic dynamics the rejuvenation of the geometric theory of dynamical systems. He is one of the pioneers in the introduction of infinite-dimensional manifolds for the study of nonlinear problems in the calculus of variations and partial differential equations. The list goes on: the systematic use of differential techniques in microeconomics, electrical circuit theory, chaos in predator–prey equations and, finally, for the twentieth century, the foundations of computational mathematics and complexity theory, and now, in the twenty-first century, the theory of learning. It has been our privilege to be among his collaborators and students in the broadest sense of the word. With these issues (Volume 5 Number 4 and Volume 6 Number 1, as well as an earlier article appearing in Volume 5 Number 2, are dedicated to Steve Smale’s 75th Birthday) we celebrate Steve’s 75th birthday and continuing vitality. He sets the bar high. We do our best.     

Existence Theorems and Estimates of Solutions for Equations of Principal Resonance

In this paper, we study systems of nonlinear, nonautonomous, ordinary differential equations that appear in the theory of averaging of nonlinear oscillations. We prove existence theorems for them and obtain conditions under which variables of the type of amplitude (or energy) are uniformly bounded with respect to time.     

Differential equations with bounded positive Green’s functions and generalized Aizerman’s hypothesis

We derive conditions for the positivity and boundedness of the Green functions of the higher order linear nonautonomous ODE. By virtue of these conditions, the existence of positive solutions for a class of nonlinear equations is proved. In addition, upper and lower estimates for the Green functions are established. Moreover, it is shown that nonlinear equations, having separated nonautonomous linear parts, satisfy the generalized Aizerman hypothesis on absolute stability, if they have the positive Green functions.     

Even periodic solutions of higher order duffing differential equations

By using Mawhin’s continuation theorem, the existence of even solutions with minimum positive period for a class of higher order nonlinear Duffing differential equations is studied.     

A scenario for the creation of chaotic attractors in Chua’s system

We investigate a scenario for the creation of irregular chaotic attractors in Chua’s system. We show that irregular attractors in Chua’s system are created by those and only those mechanisms that characterize Lorenz, Rössler, and other dissipative nonlinear systems described by ordinary differential equations. These mechanisms include cascades of Feigenbaum period doubling bifurcations, subharmonic cascades of cycle bifurcations in Sharkovskii’s order, and homoclinic cascades of bifurcations.