On the perron sign change effect for Lyapunov characteristic exponents of solutions of differential systems

The Perron effect is the effect in which the characteristic Lyapunov exponents of solutions of a differential system change sign from negative to positive when passing to a perturbed system. We show that this effect is realized on all nontrivial solutions of two two-dimensional systems: an original linear system with negative characteristic exponents and a perturbed system with small perturbations of arbitrary order m > 1 in a neighborhood of the origin, all of whose nontrivial solutions have positive characteristic exponents. We compute the exact positive value of the characteristic exponents of solutions of the two-dimensional nonlinear Perron system with small second-order perturbations, which realizes only a partial Perron effect.     

Continual version of the Perron effect of change of values of the characteristic exponents

We prove the existence of a perturbed two-dimensional system of ordinary differential equations such that its linear approximation has arbitrarily prescribed negative characteristic exponents, the perturbation is of arbitrarily prescribed higher order of smallness in a neighborhood of the origin, all of its nontrivial solutions are infinitely extendible to the right, and the whole set of their Lyapunov exponents is contained in the positive half-line, is bounded, and has positive Lebesgue measure. In the general case, we also obtain explicit representations of the exponents of these solutions via their initial values.     

On the Baire Classification of Positive Characteristic Exponents in the Perron Effect of Change of Their Values

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ?ⁿ {0}.     

Sign reversal of the characteristic exponents of solutions of a differential system with initial data on finitely many points and lines

We realize a version of the Perron sign reversal effect for the characteristic exponents of a two-dimensional differential system; the exponents are negative for the linear approximation system and positive for the nontrivial solutions of the full nonlinear system with a higher-order perturbation in a neighborhood of the origin and with initial data on an arbitrary finite set of points and lines on the plane R ².     

Multidimensional analog of the two-dimensional perron effect of sign change of characteristic exponents for infinitely differentiable differential systems

We obtain a general n-dimensional analog of the two-dimensional (partial) Perron effect of sign change of all arbitrarily prescribed negative characteristic exponents of an n-dimensional differential system of the linear approximation with infinitely differentiable bounded coefficients to the positive sign for the characteristic exponents of all nontrivial solutions of a nonlinear n-dimensional differential system with infinitely differentiable perturbations of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. These positive exponents take n values distributed over n arbitrarily prescribed disjoint intervals and are realized on solutions issuing from nested subspaces R ¹ ? R ² ? ... ? R ⁿ .     

Lyapunov numbers of markov solutions of linear stochastic systems

All nontrivial solutions of x = A(t)x grow exponentially with rate X(x,w)e{A1,...,Xr}, A a (strictly) stationary matrix process. Projecting x to the unit sphere one obtains for each of the Lyapunov exponents Xt a solution xt with stationary angle st. Now if A is a Markov process one can restrict oneself to Markov solutions, i.e., (x, A) shall be a (joint) Markov process (wich is a restriction on the inital conditions). We prove that whenever there is a Markov solution x with Lyapunov number X then there is another Markov solution with a stationary angle (or equivalently: an invariant measure for the transition probabilities of (s, A)) with the same Lyapunov number. This has some consequences, e.g., for the uniqueness of the Lyapunov numbers     

The existence of positive solution of elliptic system by a linking theorem on product space

This paper is devoted to study the existence of nontrivial positive solution of a class of elliptic system with Dirichlet Data. By using the abstract linking theorems on product space we established in Zhao et al. (Nonlinear Anal. 49 (2002) 431) we obtain the existence of three nonnegative solutions for a class of elliptic systems and the existence of a nontrivial positive solution for the problem related to the model of competing species systems involving critical Sobolev exponents.     

A formula for the lyapunov numbers of a stochastic flow. application to a perturbation theorem

We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system. (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow). This formula is analogous to that of Khas'minskii, who deals with a linear system. We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (i.e. all the exponents are negative) then so are certain stochastic perturbations of it.     

含临界指数与Hardy项耦合椭圆系统的非平凡解

研究了一类含临界指数耦合非线性项的奇异椭圆方程组,通过对临界耦合非线性项的分析与精确的能量估计,利用环绕定理,得到了这类方程组非平凡解的存在性.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, 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. 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 Poincaré 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 Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

Lyapunov exponent,Liao perturbation and persistence Dedicated to the Memory of Professor Shantao Liao at the Centenary of His Birth

Consider a C~1 vector field together with an ergodic invariant probability that has ? nonzero Lyapunov exponents. Using orthonormal moving frames along a generic orbit we construct a linear system of ?differential equations which is a linearized Liao standard system. We show that Lyapunov exponents of this linear system coincide with all the nonzero exponents of the given vector field with respect to the given ergodic probability. Moreover, we prove that these Lyapunov exponents have a persistence property meaning that a small perturbation to the linear system(Liao perturbation) preserves both the sign and the value of the nonzero Lyapunov exponents.     

A degenerate elliptic system with variable exponents

We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial weak solutions of the system. Several consequences of the main theorem are derived; in particular, the existence of at lease two distinct nontrivial non-negative solutions is established for a scalar degenerate problem. One example is provided to show the applicability of our results.     

奇异扰动的p-Laplace方程非负非平凡解和正解的结构

精确地刻画了某些奇异扰动的p-Laplace方程非负非平凡解和正解的结构．利用上下解方法证明，方程存在很多非负非平凡的尖峰解和正的过渡尖峰解．当参数充分小时还对每个尖峰解支集的上下界进行了估计．     

拟线性椭圆型方程的非线性边值问题——Sobolev临界指数的情形

本文讨论一类具有Ｓｏｂｏｌｅｖ临界指数的拟线性椭圆型方程非线性边值问题的非平凡解的存在性,利用集中紧原理得到一个存在性结果。     

Finite-dimensional Perron effect of change of all values of characteristic exponents of differential systems

We obtain a finite-dimensional Perron effect of change of values λ ₁ ≤ … ≤ λ _n < 0 of all arbitrarily specified negative characteristic exponents of the n-dimensional system of linear approximation with infinitely differentiable bounded coefficients to arbitrarily specified, arranged in ascending order, values β _k ≥ λ _k , k = 1, …, n, of characteristic exponents of all nontrivial solutions of an n-dimensional nonlinear differential system with an infinitely differentiable perturbation of arbitrary order m > 1 of smallness in a neighborhood of the origin and growth outside it. Each value β _k is realized by all nontrivial solutions of the perturbed system issuing from the difference R ^k |R ^k?1 of embedded subspaces R ¹ ? R ² ? … ? R ⁿ .     

Loss of positivity in a nonlinear scalar heat equation

Using the theory of fixed point index, we discuss the existence of nontrivial (multiple) solutions of a nonlinear scalar heat equation with nonlocal boundary conditions depending on a positive parameter. Solutions lose positivity as the parameter decreases. For a certain parameter range, not all solutions can be positive but there are positive solutions for certain types of nonlinearity. We also prove a uniqueness result.     

Stochastic virus dynamics with Beddington-DeAngelis functional response

Stochastic virus dynamics modeled by a system of stochastic differential equations with Beddington-DeAngelis functional response and driven by white noise is investigated. The global existence of positive solutions and the existence of stationary distribution are proved. Upper and lower bounds of the pathwise and asymptotic moments for the positive solutions are sharply estimated. The absorbing property in time average is shown and the moment Lyapunov exponents are proved to be nonpositive.     

Multiplicity of positive solutions to weighted nonlinear elliptic system involving critical exponents

This paper deals with the existence and multiplicity of nontrivial solutions to a weighted nonlinear elliptic system with nonlinear homogeneous boundary condition in a bounded domain. By using the Caffarelli-Kohn-Nirenberg inequality and variational method, we prove that the system has at least two nontrivial solutions when the parameter λ belongs to a certain subset of R.     

Nonexistence results for some nonlinear nonlocal elliptic inequalities with variable exponents

We provide sufficient conditions for the nonexistence of nontrivial nonnegative solutions for some nonlinear elliptic inequalities involving the fractional Laplace operator and variable exponents. The used techniques are based on the test function method. Copyright © 2016 John Wiley & Sons, Ltd.     

A new hyper-chaotic system and its synchronization

This paper presents a new hyper-chaotic system obtained by adding a nonlinear controller to the third equation of the three-dimensional autonomous Chen–Lee chaotic system. Computer simulations demonstrated the hyper-chaotic dynamic behaviors of the system. Numerical results revealed that the new hyper-chaotic system possesses two positive exponents. It was also found that the structure of the hyper-chaotic attractors is more complex than those of the Chen–Lee chaotic system. Furthermore, the hybrid projective synchronization (HPS) of the new hyper-chaotic systems was studied using a nonlinear feedback control. The nonlinear controller was designed according to Lyapunov’s direct method to guarantee HPS, which includes synchronization, anti-synchronization, and projective synchronization. Numerical examples are presented in order to illustrate HPS.