Box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems

A class of two-dimensional linear differential systems is considered. The box-counting dimension of the graphs of solution curves is calculated. Criteria to obtain the box-counting dimension of spirals are also established.Comment: 20 pages, 6 figure

Approximation of limit cycle of differential systems with variable coefficients

summary:The behavior of the approximate solutions of two-dimensional nonlinear differential systems with variable coefficients is considered. Using a property of the approximate solution, so called conditional Ulam stability of a generalized logistic equation, the behavior of the approximate solution of the system is investigated. The obtained result explicitly presents the error between the limit cycle and its approximation. Some examples are presented with numerical simulations

Box-counting dimension of solution curves for a class of two-dimensional nonautonomous linear differential systems

A class of two-dimensional linear differential system is considered. The box-counting dimension of the graphs of solution curves is calculated. Criteria to obtain the box-counting dimension of spirals are also established

Best Ulam constants for two-dimensional non-autonomous linear differential systems

This study deals with the Ulam stability of non-autonomous linear differential systems without assuming the condition that they admit an exponential dichotomy. In particular, the best (minimal) Ulam constants for two-dimensional non-autonomous linear differential systems with generalized Jordan normal forms are derived. The obtained results are applicable not only to systems with solutions that exist globally on $(-\infty,\infty)$, but also to systems with solutions that blow up in finite time. New results are included even for constant coefficients. A wealth of examples are presented, and approximations of node, saddle, and focus are proposed. In addition, this is the first study to derive the best Ulam constants for non-autonomous systems other than periodic systems.Comment: 37 pages and 3 figure

Best Ulam constants for damped linear oscillators with variable coefficients

This study uses an associated Riccati equation to study the Ulam stability of non-autonomous linear differential vector equations that model the damped linear oscillator. In particular, the best (minimal) Ulam constants for these non-autonomous linear differential vector equations are derived. These robust results apply to vector equations with solutions that blow up in finite time, as well as to vector equations with solutions that exist globally on $(-\infty,\infty)$. Illustrative, non-trivial examples are presented, highlighting the main results.Comment: 22 page

Rectifiability of orbits for two-dimensional nonautonomous differential systems

The present study is concerned with the rectifiability of orbits for the twodimensional nonautonomous differential systems. Criteria are given whether the orbit has a finite length (rectifiable) or not (nonrectifiable). The global attractivity of the zero solution is also discussed. In the linear case, a necessary and sufficient condition can be obtained. Some examples and numerical simulations are presented to explain the results

Global asymptotic stability for half-linear differential systems with coefficients of indefinite sign

summary:This paper is concerned with the global asymptotic stability of the zero solution of the half-linear differential system $$x^{\prime } = -\,e(t)x + f(t)\phi _{p^*}\!(y)\,,\quad y^{\prime } = -\,g(t)\phi _p(x) - h(t)y\,,$$ where $p > 1$, $p^* > 1$ ($1/p + 1/p^* = 1$), and $\phi _q(z) = |z|^{q-2}z$ for $q = p$ or $q = p^*$. The coefficients are not assumed to be positive. This system includes the linear differential system $\mathbf{x}^{\prime } = A(t)\mathbf{x}$ with $A(t)$ being a $2 \times 2$ matrix as a special case. Our results are new even in the linear case ($p = p^*\! = 2$). Our results also answer the question whether the zero solution of the linear system is asymptotically stable even when Coppel’s condition does not hold and the real part of every eigenvalue of $A(t)$ is not always negative for $t$ sufficiently large. Some suitable examples are included to illustrate our results

Integral averaging technique for oscillation of damped half-linear oscillators

summary:This paper is concerned with the oscillatory behavior of the damped half-linear oscillator $(a(t)\phi _p(x'))'+b(t)\phi _p(x')+c(t)\phi _p(x) = 0$, where $\phi _p(x) = |x|^{p-1}\mathop {\rm sgn} x$ for $x \in \mathbb {R}$ and $p > 1$. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young's inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if $p \neq 2$ is presented