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Further Qualitative Results for Second-Order Dynamical Systems Based on Circuit Theory Approach

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Abstract

In this study, our goal is to construct mathematical models of some physical systems and then determine their qualitative behaviors by Lyapunov’s direct method. The constructed systems are governed by second-order vector ordinary differential equations. Our main system involves three nonlinear elements that generalize the previously studied systems. For each system, we construct the associated energy or storage or Lyapunov function from the physical implication of the system which is based on basic circuit theory. We use the power-energy relationship to construct the Lyapunov function for each system. The directional derivative of each function gives the negative value of the dissipative power in the system. These two technics may not be clear in the literature. Thus, the proposed approach simplifies the derivative of Lyapunov functions which are written in integration and improves some well-known results. For a physical system, we insist that the Lyapunov function and its directional derivate are unique. Our discussion includes four new results associated with the stability theory of dynamical systems. We also give a new passivity result for our main system. Finally, we give an example with simulations to elucidate the theoretical results.

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Authors and Affiliations

  1. Electricity and Energy Department, Başkale Vocational School of Higher Education, Van, Turkey

    Muhammet Ateş

  2. Department of Electric-Electronic Engineering, Siirt University, Siirt, Turkey

    Mehmet Recep Minaz

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  1. Muhammet Ateş
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Ateş, M., Minaz, M.R. Further Qualitative Results for Second-Order Dynamical Systems Based on Circuit Theory Approach. Circuits Syst Signal Process 41, 4755–4774 (2022). https://doi.org/10.1007/s00034-022-02027-1

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