Abstract
In many emerging scientific applications, the input of random initial time is often required for predicting and controlling the dynamics of real-order systems that include or does not include time-delays. In this work, we provide a new design for a class of random initial-time non-autonomous linear Caputo-type real-order time-delay systems that involves constant discrete delays. We first introduce a new comparison principle and a linear comparison lemma for Caputo derivative. By using the idea of comparison methodology and generalized Laplace transform, we develop some new elementary asymptotic theories and establish order-dependent and delay-independent conditions that give convergence of nontrivial solutions to such a class of system. One of the major challenges encountered in the investigation is discovering some simpler classes of autonomous linear real-order systems in bounding the coefficient matrices of a designed class of systems above by constant matrices in our assumptions. These assumptions including newly introduced matrices \(\Lambda \), \(\Theta _{k}\) and G provide some fundamental key tools for an effective asymptotic analysis of newly designed class of systems. A typical theorem that we develop says that if all the diagonal entries of \(\Lambda \) become negative and every root of the characteristic equation associated with the matrix \(\Lambda +\sum \nolimits _{k=1}^{m}\Theta _{k}G\) lies in the open left-half complex plane, then it is possible to predict the limiting behavior of the designed class of systems. For a practical application of interest, we show how to synchronize the possible chaotic dynamics of the coupled nonlinear Duffing oscillators if the system starts at a negative initial time through some proposed results.
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Acknowledgements
Bichitra Kumar Lenka acknowledges Indian Institute of Technology Guwahati for providing facilities for carrying out this research. The authors would like to thank the esteemed Reviewer for the insightful review which have helped us to improve the quality of the manuscript to its current form. The Associate Editor Prof. Vasily E. Tarasov and Editor in Chief Prof. José Eduardo Souza de Cursi are thanked for allowing a revision.
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Lenka, B.K., Bora, S.N. Limiting behaviour of non-autonomous Caputo-type time-delay systems and initial-time on the real number line. Comp. Appl. Math. 42, 313 (2023). https://doi.org/10.1007/s40314-023-02459-8
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DOI: https://doi.org/10.1007/s40314-023-02459-8
Keywords
- Asymptotic stability
- Caputo-type time-delay system
- Delayed Duffing oscillator
- Initial-time
- Non-autonomous real-order system
- Nonlinear system