Erratum to: Local stable manifold theorem for fractional systems

Nonlinear Dynamics, Mar 2017

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Erratum to: Local stable manifold theorem for fractional systems

Erratum to: Nonlinear Dyn Erratum to: Local stable manifold theorem for fractional systems Amey Deshpande 0 Varsha Daftardar-Gejji 0 E p 0 (t p J j ) = t 0 B j 0 C j 0 hence 0 0 A. Deshpande Department of Mathematics, College of Engineering Pune , Pune 411005 , India where B j (t ) and C j (t ) are n j × n j matrices defined as - E p,β (t p A)n×n = t 1−β B(t ) + C (t ), ” is incorrect. The corrected version of the above expression is presented below. Note that Lemma 4 part 1 remains as it is. For the corrected expression, we introduce the following notations. Notation: Let Iγ denote n j × n j matrix with entries [Iγ ]i,k = δi+γ,k , (γ = 0, . . . , n j − 1, 1 ≤ i, k ≤ n j ). Lemma 4 (Part 2 (Corrected)) For 0 < p < 1, j = 1, 2 . . . , l and q ∈ N\{1}, E p, p(t p J j ) = t − p B j (t ) + C j (t ), B j (t ) := 0 j = 1, 2, . . . , s, B j (t ) := C j (t ) := (t, λ j )In j −1 j = 1, 2, . . . , l, 1 ∂m+1 1 ψm (t, λ j ) = m! ∂λm+1 exp t λ jp k=2 q (−1)m+2(k + m)! λ −jk−m−1t − pk (k − 1)! (1 − pk) + O(|λ j |−q−m t − p− pq ) . Let B(t ) and C (t ) denote the block diagonal matrices consisting of B j (t ) and C j (t ) on the diagonal, respectively. Hence E p, p(t p A) = t − p B(t ) + C (t ). As a consequence of the above correction, appropriate changes should be made in expressions of Lemma 5, 6, 8 and the step II of the proof of the theorem. Thus in view of these corrections the proof of the local stable manifold theorem given by us in the original article continues to hold true. For details we refer the readers to our article [1]. 1. Deshpande , A. , Daftardar-Gejji , V. : Local Stable Manifold theorem for fractional systems revisited . ArXiv e-prints ( 2017 ). http://adsabs.harvard.edu/abs/ 2017arXiv170100076D


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Erratum to: Local stable manifold theorem for fractional systems, Nonlinear Dynamics, 2017, DOI: 10.1007/s11071-017-3352-1