Successive estimations of bilateral bounds and trapping/stability regions of solution to some nonlinear nonautonomous systems

Abstract

Estimation of the degree of stability and the bounds of solutions to nonautonomous nonlinear systems present major concerns in numerous applied problems. Yet, current techniques frequently yield overconservative conditions which are unable to effectively gage these characteristics in time-varying nonlinear systems. This paper develops a novel methodology providing successive approximations to solutions that are stemmed from the trapping/stability regions of these systems and estimate the errors of such approximations. In turn, this leads to successive approximations of both the bilateral bounds of solutions and the boundaries of trapping/stability regions of the underlying systems. Along these lines we formulate enhanced stability/boundedness criteria and contrast our inferences with inclusive simulations which reveal dependence of the trapping/stability regions upon the structure of time-dependent components and initial time moment.

Appendix

Let us sketch, for instance, how to derive inequalities (7.2). Assume that $f\left( {t,x} \right) = [f_{1} , \ldots ,f_{n} ]^{\text{T}}$, $x,\,\,u\,,z \in {\mathbb{R}}^{n}$, $f_{i} \left( {t,x} \right) = \zeta_{i} \left( t \right)x_{i}^{{d_{i} }} = \zeta_{i} \left( t \right)\left( {z_{i} + u_{i} } \right)^{{d_{i} }} ,\,\,d_{i} \in {\mathbb{N}}$. Thus, $\left| {f_{i} } \right| = \left| {\zeta_{i} \left( {z_{i} + u_{i} } \right)^{{d_{i} }} } \right| \le \left| {\zeta_{i} } \right|\left| {z_{i} + u_{i} } \right|^{{d_{i} }}$$\le \left| {\zeta_{i} } \right|\left( {\left| {z_{i} } \right| + \left| {u_{i} } \right|} \right)^{{d_{i} }} \le \left| {\zeta_{i} } \right|\left( {\left\| z \right\| + \left| {u_{i} } \right|} \right)^{{d_{i} }}$$\le \left| {\zeta_{i} } \right|\left( {\left\| z \right\| + \left\| u \right\|} \right)^{{d_{i} }}$.Then, taking into account that $\left\| f \right\| \le \left\| f \right\|_{1} = \sum\nolimits_{i = 1}^{n} {\left| {f_{i} } \right|}$, we infer that $\gamma_{*} \left( {t,u} \right) = \sum\nolimits_{i = 1}^{n} {\left| {\zeta_{i} } \right|\left| {u_{i} } \right|^{{d_{i} }} }$, $\varPi_{*} \left( {t,y} \right) = \,\sum\nolimits_{i = 1}^{n} {\left| {\zeta_{i} } \right|\left( {\left\| z \right\| + \left| {u_{i} } \right|} \right)^{{d_{i} }} } - \gamma_{*} \left( {t,u} \right)$, $\hat{\gamma }\left( {t,\left\| u \right\|} \right) = \sum\nolimits_{i = 1}^{n} {\left| {\zeta_{i} } \right|\left\| u \right\|^{{d_{i} }} }$ if $F_{0} = 0$. Subsequently, we enter in this formula that $\left\| u \right\| \le \left\| u \right\|_{1}$. The prior formula holds if $F_{0} > 0$ with obvious change in notation.

Assume next that $f_{id} \left( {t,x} \right) = \zeta_{id} \left( t \right)x^{d}$, where $x^{d} = x_{1}^{{d_{1} }} \ldots x_{n}^{{d_{n} }} ,\,\,d_{i} \in {\mathbb{Z}},\,\,\,d = \sum\nolimits_{i = 1}^{n} {d_{i} }$. Then, $\left| {f_{id} \left( {t,x} \right)} \right| = \left| {\zeta_{id} } \right|\left| {x_{1}^{{d_{1} }} } \right| \ldots \left| {x_{n}^{{d_{n} }} } \right|$ which enables utility of the prior procedure for every $\left| {x_{i}^{{d_{i} }} } \right|$. Clearly, such sequence of steps yields (7.2) if $f\left( {t,x} \right)$ is a vector polynomial in $x$.