This technical note is concerned with a maximum principle for a new class of non-zero sum stochastic differential games. The most distinguishing feature, compared with the existing literature, is that the game systems are described by backward stochastic differential equations (BSDEs). This kind of games are motivated by some interesting phenomena arising from financial markets and can be used to characterize the players with different levels of utilities. We establish a necessary condition and a sufficient condition in the form of maximum principle for open-loop equilibrium point of the foregoing games respectively. To explain the theoretical results, we use them to study a financial problem.

A Pontryagin's Maximum Principle for Non-Zero Sum Differential Games of BSDEs with Applications

This paper presents chaos synchronization between two different chaotic systems by using a nonlinear controller, in which the nonlinear functions of the system are used as a nonlinear feedback term. The feedback controller is designed on the basis of stability theory, and the area of feedback gain is determined. The artificial simulation results show that this control method is commendably effective and feasible.

http://iopscience.iop.org/article/10.1088/1009-1963/16/6/019/pdf

Synchronization between two different chaotic systems with nonlinear feedback control

https://opensiuc.lib.siu.edu/cgi/viewcontent.cgi?article=1116&context=math_articles

State and Feedback Linearizations of Single-Input Control Systems

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincare maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincare sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

The phenomenon of length-scale competition in soliton-bearing equations perturbed by spatially dependent terms [A. Sanchez and A. R. Bishop, SIAM Rev. 40, 579 (1998)] is analyzed from a general viewpoint. We show that the perturbation gives rise to an effective potential for solitons, which consists of wells and barriers. We calculate the effect of these potential features on the solitons, establishing a direct relationship between the maxima, minima, and curvature of the potential with soliton deformations. When the typical wavelength of the perturbation is of the order of the soliton width, these deformations are seen to correspond to the excitation of shape modes and can lead to the dissipation of the soliton kinetic energy and, further, to the impossibility of soliton propagation. Thus, we demonstrate that the mechanism for length-scale competition is related to changes in the dynamics of the internal modes. We study different examples where the perturbation is introduced parametrically and nonparametrically to make it clear that our results apply to a wide class of equations.

/pdf/kink-dynamics-in-spatially-inhomogeneous-media-the-role-of-3bzbhiobq1.pdf

Kink dynamics in spatially inhomogeneous media: the role of internal modes.

Controlling chaos in the Lorenz system with a controllable Rayleigh number is investigated by the state space exact linearization method. Based on proving the exact linearizability, the nonlinear feedback is utilized to design the transformation changing the original chaotic system into a linear controllable one so that the control is realized. Numerical examples of control are presented.

Control of the lorenz chaos by the exact linearization

A cross section of the rod is taken as object of investigation The freedom of the section in free or constraint case is analyzed and the definition of virtual displacement of the section is given, which can be expressed by a variational operation Assuming the variational and partial differential operations has commutativity, based on the hypothesis about surface constraint subjected to the rod, the freedom of the section on constraint surface is discussed and the equations satisfied by virtual displacements of the section are given Combining D'Alembert principle and the principle of virtual work, D'Alembert-Lagrange principle is established When constitutive equation of material of the rod is linear, the principle can be transformed to Euler-Lagrange form From the principle, a dynamical equation in various forms such as Kirchhoff, Lagrange, Nielsen and Appell equation can be derived For the case when a rod is subjected to a surface or a nonholonomic constraint, Lagrange equation with undetermined multipliers is obtained Integral variational principle of dynamics of a super-thin elastic rod is also established, from which Hamilton principle formulation is obtained when the material of the rod is linear Finally, canonical variables to describe the state of the section and Hamilton function are defined, and Hamilton canonical equation is derived The analytical methods of dynamical modeling of a super-thin elastic rod have been constructed, which can serve as a theoretical framework of analytical dynamics of a super-thin elastic rod with two independent variables

Methods of analytical mechanics for dynamics of the Kirchhoff elastic rod

This paper deals with the chaotic attitude motion of a magnetic rigid spacecraft with internal damping in an elliptic orbit. The dynamical model of the spacecraft is established. The Melnikov analysis is carried out to prove the existence of a complicated nonwandering Cantor set. The dynamical behaviors are numerically investigated by means of time history, Poincare map, Lyapunov exponents and power spectrum. Numerical simulations demonstrate the chaotic motion of the system. The input-output feedback linearization method and its modified version are applied, respectively, to control the chaotic attitude motions to the given fixed point or periodic motion.

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

The synchronization of two symmetrical nonlinear-coupled chaotic systems is discussed. A special nonlinear-coupled term is constructed by suitable separation between linear and nonlinear terms of the chaotic system. The phenomenon of stab le chaotic synchronization has been found in a certain region of the coupled str ength α=05. The stability of the synchronous state is examined by the sta bility criterion of linear systems and the conditional Lyapunov exponent. The Lo renz system and the hyper-chaotic Rssler system are treated as numerical exam ples. It is shown that the proposed method is effective for chaotic synchronizat ion of time-continued systems and hyper-chaotic systems.

Synchronization of symmetrically nonlinear-coupled chaotic systems

The extended Schrodinger equation for the Kirchhoff elastic rod with noncircular cross section is derived using the concept of complex rigidity. As a mathematical model of supercoiled DNA, the Schrodinger equation for the rod with circular cross section is a special case of the equation derived in this paper. In the twistless case of the rod when the principal axes of the cross section are coincident with the Frenet coordinates of the centreline, the Schrodinger equation is transformed to the Duffing equation. The equilibrium and stability of the twistless rod are discussed and a bifurcation phenomenon is presented.

Liu Yanzhu

Papers

Control of the lorenz chaos by the exact linearization

Methods of analytical mechanics for dynamics of the Kirchhoff elastic rod

Chaotic attitude motion of a magnetic rigid spacecraft in an elliptic orbit and its control

Synchronization of symmetrically nonlinear-coupled chaotic systems

The Schr?dinger equation for a Kirchhoff elastic rod with noncircular cross section