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The Shil\u2019nikov lemma provides a useful theoretical tool to prove the existence of chaos in three-dimensional smooth autonomous systems. It requires, however, the proof of existence of a homoclinic or heteroclinic orbit, which remains a very difficult technical problem if contigent on data. In this paper, for the Chen system with linear time-delay feedback, we demonstrate a homoclinic orbit by using a modified undetermined coefficient method and we propose a spiral involute projection method. In such a way, we identify experimentally the asymmetrical homoclinic orbit in order to apply the Shil\u2019nikov-type lemma and to show that chaos is indeed generated in the Chen circuit with linear time-delay feedback. We also identify the presence of a single-scroll attractor in the Chen system with linear time-delay feedback in our experiments. We confirm that the Chen single-scroll attractor is hyperchaotic by numerically estimating the finite-time local Lyapunov exponent spectrum. By means of a linear scaling in the coordinates and the time, such a method can also be applied to the generalized Lorenz-like systems. The contribution of this work lies in: first, we treat the trajectories corresponding to the real eigenvalue and the image eigenvalues in different ways, which is compatible with the characteristics of the trajectory geometry; second, we propose a spiral involute projection method to exhibit the trajectory corresponding to the image eigenvalues; third, we verify the homoclinic orbit by experimental data.<\/jats:p>","DOI":"10.1142\/s0218127419501141","type":"journal-article","created":{"date-parts":[[2019,8,27]],"date-time":"2019-08-27T11:09:58Z","timestamp":1566904198000},"page":"1950114","source":"Crossref","is-referenced-by-count":12,"title":["Existence of Chaos in the Chen System with Linear Time-Delay Feedback"],"prefix":"10.1142","volume":"29","author":[{"given":"Kun","family":"Tian","sequence":"first","affiliation":[{"name":"Shaanxi Key Laboratory of Complex System Control and Intelligent Information Processing, Xi\u2019an University of Technology, Xi\u2019an 710048, P. R. 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