The notions of mixed monotone decomposition of dynamical systems are introduced. The fundamental idea is to make an elaborate use of the natural growth and decay mechanism inherent in the structure of a dynamical syst...The notions of mixed monotone decomposition of dynamical systems are introduced. The fundamental idea is to make an elaborate use of the natural growth and decay mechanism inherent in the structure of a dynamical systems to decompose its dynamics into increase and decrease parts, and thereby to constitute an augmented dynamical system as the so_called "two_sided comparison system" of the original one. The corresponding two_sided comparison theorems show that the solution of the comparison system gives lower and upper bounds of that of the original system. Therefore, the properties of a dynamical system can be obtained through the study of its two_sided comparison system. Compared with the conventional comparison method in literature, the mixed monotone decomposition method developed herein takes in a natural way structural properties of dynamical systems into account instead of requiring strict conditions of (quasi_)monotonicity on them, and could yields refined, usually nonsymmetrical, state estimates, and thus is more suitable for systems with nonsymmetrical state constraints. As an application of the method, a sufficient condition is established for the global asymptotic stability of the trivial solution of a class of continuous_time systems with nonsymmetrical state saturation. The condition is given in terms of coefficients and state saturation levels of such systems, and contains as a special case a recent result on systems with symmetric state saturation in literature.展开更多
Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As ...Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature （possibly strictly positive on some sections）, whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.展开更多
文摘The notions of mixed monotone decomposition of dynamical systems are introduced. The fundamental idea is to make an elaborate use of the natural growth and decay mechanism inherent in the structure of a dynamical systems to decompose its dynamics into increase and decrease parts, and thereby to constitute an augmented dynamical system as the so_called "two_sided comparison system" of the original one. The corresponding two_sided comparison theorems show that the solution of the comparison system gives lower and upper bounds of that of the original system. Therefore, the properties of a dynamical system can be obtained through the study of its two_sided comparison system. Compared with the conventional comparison method in literature, the mixed monotone decomposition method developed herein takes in a natural way structural properties of dynamical systems into account instead of requiring strict conditions of (quasi_)monotonicity on them, and could yields refined, usually nonsymmetrical, state estimates, and thus is more suitable for systems with nonsymmetrical state constraints. As an application of the method, a sufficient condition is established for the global asymptotic stability of the trivial solution of a class of continuous_time systems with nonsymmetrical state saturation. The condition is given in terms of coefficients and state saturation levels of such systems, and contains as a special case a recent result on systems with symmetric state saturation in literature.
基金supported by National Natural Science Foundation of China (Grant No. 11231001)Education Ministry of China
文摘Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature （possibly strictly positive on some sections）, whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.
