Construction of a CPA contraction metric for periodic orbits using semidefinite optimization

A Riemannian metric with a local contraction property can be used to prove existence and uniqueness of a periodic orbit and determine a subset of its basin of attraction. While the existence of such a contraction metric is equivalent to the existence of an exponentially stable periodic orbit, the explicit construction of the metric is a difficult problem.In this paper, the construction of such a contraction metric is achieved by formulating it as an equivalent problem, namely a feasibility problem in semidefinite optimization. The contraction metric, a matrix-valued function, is constructed as a continuous piecewise affine (CPA) function, which is affine on each simplex of a triangulation of the phase space. The contraction conditions are formulated as conditions on the values at the vertices.The paper states a semidefinite optimization problem. We prove on the one hand that a feasible solution of the optimization problem determines a CPA contraction metric and on the other hand that the optimization problem is always feasible if the system has an exponentially stable periodic orbit and the triangulation is fine enough. An objective function can be used to obtain a bound on the largest Floquet exponent of the periodic orbit.     

Borg’s criterion for almost periodic differential equations

Borg’s criterion is used to prove the existence of an exponentially asymptotically stable periodic orbit of an autonomous differential equation and to determine its domain of attraction. In this article, this method is generalized to almost periodic differential equations. Both sufficient and necessary conditions are obtained for the existence of an exponentially stable almost periodic solution. The condition uses a Riemannian metric, and an example for the explicit construction of such a metric is presented.     

On Quasi-weakly Almost Periodic Points of Continuous Flows

Let X be a compact metric space, F : X ×R→ X be a continuous flow and x ∈ X a proper quasi-weakly almost periodic point, that is, x is quasi-weakly almost periodic but not weakly almost periodic. The aim of this paper is to investigate whether there exists an invariant measure generated by the orbit of x such that the support of this measure coincides with the minimal center of attraction of x? In order to solve the problem, two continuous flows are constructed. In one continuous flow,there exist a proper quasi-weakly almost periodic point and an invariant measure generated by its orbit such that the support of this measure coincides with its minimal center of attraction; and in the other,there is a proper quasi-weakly almost periodic point such that the support of any invariant measure generated by its orbit is properly contained in its minimal center of attraction. So the mentioned problem is sufficiently answered in the paper.     

Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions

The basin of attraction of an asymptotically stable fixed point of the discrete dynamical system given by the iteration x_n+1=g(x_n) can be determined through sublevel sets of a Lyapunov function. In Giesl [On the determination of the basin of attraction of discrete dynamical systems. J. Difference Equ. Appl. 13(6) (2007) 523–546] a Lyapunov function is constructed by approximating the solution of a difference equation using radial basis functions. However, the resulting Lyapunov function is non-local, i.e. it has no negative discrete orbital derivative in a neighborhood of the fixed point. In this paper we modify the construction method by using the Taylor polynomial and thus obtain a Lyapunov function with negative discrete orbital derivative both locally and globally.     

Approximation on attraction domain of Cohen-Grossberg neural networks

In this paper, approximations of attraction domains of the asymptotically stable equilibrium points of some typical Cohen-Grossberg neural networks are achieved. Most Cohen-Grossberg neural networks are highly nonlinear systems which makes it difficult to approximate their attraction domain. Under some weak assumptions, we are allowed to employ the optimal Lyapunov method to obtain a Lyapunov function for asymptotically stable equilibrium points of a given Cohen-Grossberg neural network. With the help of this Lyapunov function, we approximate the corresponding attraction domain by the iterative expansion approach. Numerical simulations also illustrate that the approximation obtained is really part of the attraction domain.     

Revised CPA method to compute Lyapunov functions for nonlinear systems

The CPA method uses linear programming to compute Continuous and Piecewise Affine Lyapunov functions for nonlinear systems with asymptotically stable equilibria. In [14] it was shown that the method always succeeds in computing a CPA Lyapunov function for such a system. The size of the domain of the computed CPA Lyapunov function is only limited by the equilibrium?s basin of attraction. However, for some systems, an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori. This is necessary, if the equilibrium is not exponentially stable, because the existence of a CPA Lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in [11]. However, if the equilibrium is exponentially stable, then this was an artifact of the method. In this paper we overcome this artifact by developing a revised CPA method. We show that this revised method is always able to compute a CPA Lyapunov function for a system with an exponentially stable equilibrium. The only conditions on the system are that it is C²$C^2$ and autonomous. The domain of the CPA Lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction. Whereas in a previous paper [10] we have shown these results for planar systems, in this paper we cover general n-dimensional systems.     

Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust

This article deals with the optimal transfer of a satellite between Keplerian orbits using low propulsion and is based on preliminary results of Epenoy et al. (1997) where the optimal trajectories of the energy minimization problem are approximated using averaging techniques. The averaged Hamiltonian system is explicitly computed. It is related to a Riemannian problem whose distance is an approximation of the value function. The extremal curves are analyzed, proving that the system remains integrable in the coplanar case. It is also checked that the metric associated with coplanar transfers towards a circular orbit is flat. Smoothness of small Riemannian spheres ensures global optimality of the computed extremals.     

一类三阶非线性系统的全局渐近稳定性

对一类三阶非线性系统构造出了较好的Lyapunov函数,得到其零解全局渐近稳定的充分性准则,而且去掉了一般要求Lyapunov函数具有无穷大这个较强的条件,只要求系统正半轨线有界,所得结果包含并改进了旧有的结果.     

Homologically versus homotopically area minimizing surfaces in 3-manifolds

We construct a Riemannian metric on the 3-torus such that no closed surface minimizing area in its homology class is incompressible, i.e., each such surface is of genus greater than one. In particular, for such a Riemannian metric, the homotopically area minimizing 2-tori constructed in [5] do not minimize area in their homology classes. The example is easily generalized to arbitrary 3-manifolds. The constructed Riemannian metric can be chosen to be conformally equivalent to any arbitrary given one. Received September 4, 1998 / Accepted October 23, 1998     

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points.To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.     

Long periodic shadowing

A general theorem for establishing the existence of a true periodic orbit near a numerically computed pseudoperiodic orbit of an autonomous system of ordinary differential equations is presented. For practical applications, a Newton method is devised to compute appropriate pseudoperiodic orbits. Then numerical considerations for checking the hypotheses of the theorem in terms of quantities which can be computed directly from the pseudoperiodic orbit and the vector field are addressed. Finally, a numerical method for estimating the Lyapunov exponents of the true periodic orbit is given. The theory and computations are designed to be applicable for unstable periodic orbits with long periods. The existence of several such periodic orbits of the Lorenz equations is exhibited. This revised version was published online in June 2006 with corrections to the Cover Date.     

Min-switching local stabilization for discrete-time switching systems with nonlinear modes

This paper deals with the discrete-time switched Lur’e problem in finite domain. The aim is to provide a stabilization inside an estimate of the origin’s basin of attraction, as large as possible, via a suitable switching rule. The design of this switching rule is based on the min-switching policy and can be induced by sufficient conditions given by Lyapunov–Metzler inequalities. Nevertheless instead of intuitively considering a switched quadratic Lyapunov function for this approach, a suitable switched Lyapunov function including the modal nonlinearities is proposed. The assumptions required to characterize the nonlinearities are only mode-dependent sector conditions, without constraints related to the slope of the nonlinearities. An optimization problem is provided to allow the maximization of the size of the basin of attraction estimate–which may be composed of disconnected sets–under the stabilization conditions. A numerical example illustrates the efficiency of our approach and emphasizes the specificities of our tools.     

On the estimation of asymptotic stability regions for autonomous nonlinear systems

The problem of computing regions of asymptotic stability forautonomous nolinear systems is reconsidered. A two-step procedureis proposed in which a suitable global Lyapunov function isfirst constructed to prove the system's nonoscillatory behaviour.Subsequently the Lyapunov function is used to compute the initialstates for a trajectory-reversing technique to estimate thesystem's stability boundaries. The method combines computationalefficiency and accuracy in obtaining a close estimate of theexact region of attraction of a stable equilibrium state.     

单隐层神经网络与最佳多项式逼近

研究单隐层神经网络逼近问题．以最佳多项式逼近为度量,用构造性方法估计单隐层神经网络逼近连续函数的速度．所获结果表明:对定义在紧集上的任何连续函数,均可以构造一个单隐层神经网络逼近该函数,并且其逼近速度不超过该函数的最佳多项式逼近的二倍．     

The metric geometry of the manifold of Riemannian metrics over a closed manifold

We prove that the L ² Riemannian metric on the manifold of all smooth Riemannian metrics on a fixed closed, finite-dimensional manifold induces a metric space structure. As the L ² metric is a weak Riemannian metric, this fact does not follow from general results. In addition, we prove several results on the exponential mapping and distance function of a weak Riemannian metric on a Hilbert/Fréchet manifold. The statements are analogous to, but weaker than, what is known in the case of a Riemannian metric on a finite-dimensional manifold or a strong Riemannian metric on a Hilbert manifold.     

一类泛函微分方程的周期解的存在性、唯一性及全局吸引性

该文研究了一类具有分布滞量的微分系统的周期解的存在性、唯一性及全局吸引性等问题.利用不动点方法和Lyapunov泛函方法,建立了保证该类系统周期解的存在性、唯一性、一致稳定性及全局吸引性的充分条件.     

Connection across a Separatrix with Dissipation

Dissipative perturbations of strongly nonlinear oscillators that correspond to slowly varying double-well potentials are considered. The method of averaging, which describes the solution as nearly periodic, fails as the trajectory approaches the unperturbed separatrix, a homoclinic orbit of the saddle point, significantly before it is captured in either well. Nevertheless, perturbed initial conditions corresponding to the boundary of the basin of attraction for each well, which are the perturbed stable manifolds of the saddle point, are accurately determined using only the method of averaging modified by Melnikov energy ideas near the separatrix. To determine the amplitude and phase of the captured oscillations after crossing the separatrix, a transition region is constructed consisting of a large sequence of nearly solitary pulses along the separatrix. The amplitude and phases of the slowly varying nonlinear oscillations away from the separatrix, both before and after capture, are matched to this transition region. In this way, analytic connection formulas across the separatrix are obtained and are shown to depend on the perturbed initial conditions.     

Cheeger-gromoll type metrics on the (1,1)-tensor bundles

Using a Riemannian metric on a differentiable manifold, a Cheeger-Gromoll type metric is introduced on the (1,1)-tensor bundle of the manifold. Then the Levi-Civita connection, Riemannian curvature tensor, Ricci tensor, scalar curvature and sectional curvature of this metric are calculated. Also, a para-Nordenian structure on the the (1,1)-tensor bundle with this metric is constructed and the geometric properties of this structure are studied.     

The Lyapunov exponents of C 1 hyperbolic systems

Let f be a C 1 diffeomorphisim of smooth Riemannian manifold and preserve a hyperbolic ergodic measure μ. We prove that if the Osledec splitting is dominated, then the Lyapunov exponents of μ can be approximated by the exponents of atomic measures on hyperbolic periodic orbits.     

Two-sided approximation to periodic solutions of ordinary differential equations

Summary A two-sided approximation to the periodic orbit of an autonomous ordinary differential equation system is considered. First some results about variational equation systems for periodic solutions are obtained in Sect. 2. Then it is proved that if the periodic orbit is convex and stable, the explicit difference solution approximates the periodic orbit from the outer part and the implicit one from the inner part respectively. Finally a numerical example is given to illustrate our result and to point out that the numerical solution no longer has a one-sided approximation property, if the periodic orbit is not convex.The Work is supported by the National Natural Science Foundation of China