Nonexistence of invariant manifolds in fractional-order dynamical systems
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ORIGINAL PAPER
Nonexistence of invariant manifolds in fractional-order dynamical systems Sachin Bhalekar
· Madhuri Patil
Received: 23 December 2019 / Accepted: 2 November 2020 © Springer Nature B.V. 2020
Abstract Invariant manifolds are important sets arising in the stability theory of dynamical systems. In this article, we take a brief review of invariant sets. We provide some results regarding the existence of invariant lines and parabolas in planar polynomial systems. We provide the conditions for the invariance of linear subspaces in fractional-order systems. Further, we provide an important result showing the nonexistence of invariant manifolds (other than linear subspaces) in fractional-order systems. Keywords Invariant manifold · Separatrix · Stability · Tangency condition · Caputo fractional derivative 1 Introduction Dynamical systems [1–4] is a trending branch of mathematics playing a vital role in the mathematical analysis as well as in the applied sciences [5–8]. Chaos theory and fractals [9–14] are the sub-branches of this theory which have attracted the attention of scientists as well as layman. The applications of dynamical systems are S. Bhalekar (B) School of Mathematics and Statistics, University of Hyderabad, Hyderabad 500046, India e-mail: [email protected] e-mail: [email protected] M. Patil Department of Mathematics, Shivaji University, Kolhapur 416004, India e-mail: [email protected]
found in arts [15,16] and social sciences [17,18] also. The theoretical results such as Hartman–Grobman theorem [4], Stable manifold theorem [4] and Poincare– Bendixson theorem [4] made substantial contributions to the mathematical analysis. In the dynamical systems theory, an invariant manifold is a set M such that every point in M is mapped to some point in the same set M under the evolution of the system. Hadamard [19], Liapunov [20] and Perron [21] proposed pioneering results in the theory of invariant manifolds. Existence and smoothness of invariant manifolds are discussed in [22]. This theory has applications in a variety of fields in science and engineering. Gorban and Karlin [23] employed this theory to reduce the description of equations arising in chemical kinetics. Beigie et al. [24] used invariant manifolds in the study of chaotic advection. Control of chaos is achieved by using invariant manifolds by Chen et al. [25]. The globally convergent, reduced-order observers are designed for general nonlinear systems in [26]. In [27], the dynamics of a nonlinear rotor is investigated using the invariant manifolds. Roussel and Fraser [28] utilized this theory to simplify the metabolic models arising in biochemistry. Fractional calculus deals with the differentiation and integration of arbitrary order [29–36]. The fractional derivative operators are non-local and hence very useful in modeling the memory in the natural systems [37– 43]. The existence and uniqueness of the solution of fractional-order initial value problems are discussed in
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[35,44–46]. Stab
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