{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,8]],"date-time":"2025-09-08T06:08:56Z","timestamp":1757311736462},"reference-count":30,"publisher":"World Scientific Pub Co Pte Lt","issue":"08","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2016,7]]},"abstract":"<jats:p> In this paper, we study the dynamics of autonomous ODE systems with [Formula: see text] symmetry. First, we consider eight weakly-coupled oscillators and establish the condition for the existence of stable heteroclinic cycles in most generic [Formula: see text]-equivariant systems. Then, we analyze the action of [Formula: see text] on [Formula: see text] and study the pattern of periodic solutions arising from Hopf bifurcation. We identify the type of periodic solutions associated with the pairs [Formula: see text] of spatiotemporal or spatial symmetries, and prove their existence by using the [Formula: see text] Theorem due to Hopf bifurcation and the [Formula: see text] symmetry. In particular, we give a rigorous proof for the existence of a fourth branch of periodic solutions in [Formula: see text]-equivariant systems. Further, we apply our theory to study a concrete case: two coupled van der Pol oscillators with [Formula: see text] symmetry. We use normal form theory to analyze the periodic solutions arising from Hopf bifurcation. Among the families of the periodic solutions, we pay particular attention to the phase-locked oscillations, each of them being embedded in one of the invariant manifolds, and identify the in-phase, completely synchronized motions. We derive their explicit expressions and analyze their stability in terms of the parameters. <\/jats:p>","DOI":"10.1142\/s0218127416501418","type":"journal-article","created":{"date-parts":[[2016,8,5]],"date-time":"2016-08-05T05:03:24Z","timestamp":1470373404000},"page":"1650141","source":"Crossref","is-referenced-by-count":3,"title":["Coupled Oscillatory Systems with \ud835\udd3b4 Symmetry and Application to van der Pol Oscillators"],"prefix":"10.1142","volume":"26","author":[{"given":"Adrian C.","family":"Murza","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Transilvania University, Bra\u015fov, Strada Iuliu Maniu 50, 500091, Bra\u015fov, Romania"}]},{"given":"Pei","family":"Yu","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Western University, London, Ontario N6A 5B7, Canada"}]}],"member":"219","published-online":{"date-parts":[[2016,8,5]]},"reference":[{"key":"S0218127416501418BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02429852"},{"key":"S0218127416501418BIB002","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100072364"},{"key":"S0218127416501418BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-011-0956-7_3"},{"key":"S0218127416501418BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-008-9402-y"},{"key":"S0218127416501418BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(98)00222-2"},{"key":"S0218127416501418BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-2789(00)00097-X"},{"key":"S0218127416501418BIB007","doi-asserted-by":"publisher","DOI":"10.1016\/j.ijsolstr.2003.11.035"},{"key":"S0218127416501418BIB008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8159-4"},{"key":"S0218127416501418BIB009","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511665639"},{"key":"S0218127416501418BIB010","doi-asserted-by":"publisher","DOI":"10.1017\/S0017089505002855"},{"key":"S0218127416501418BIB011","doi-asserted-by":"publisher","DOI":"10.1016\/j.jde.2011.09.043"},{"key":"S0218127416501418BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4574-2"},{"key":"S0218127416501418BIB013","volume-title":"The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space","author":"Golubitsky M.","year":"2003"},{"key":"S0218127416501418BIB014","doi-asserted-by":"publisher","DOI":"10.1142\/9789812567840_0001"},{"key":"S0218127416501418BIB015","volume-title":"Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields","author":"Guckenheimer J.","year":"1993","edition":"4"},{"key":"S0218127416501418BIB016","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-2918-9"},{"key":"S0218127416501418BIB017","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1996.3201"},{"key":"S0218127416501418BIB018","doi-asserted-by":"publisher","DOI":"10.1007\/BF00336943"},{"key":"S0218127416501418BIB019","doi-asserted-by":"publisher","DOI":"10.1017\/S0143385700008270"},{"key":"S0218127416501418BIB020","volume-title":"Elements of Applied Bifurcation Theory","author":"Kuznetsov Yu. 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