{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T12:53:28Z","timestamp":1747313608625},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T00:00:00Z","timestamp":1600905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Soft Comput"],"published-print":{"date-parts":[[2021,2]]},"DOI":"10.1007\/s00500-020-05325-5","type":"journal-article","created":{"date-parts":[[2020,9,24]],"date-time":"2020-09-24T12:02:39Z","timestamp":1600948959000},"page":"2521-2530","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On the dynamics: existence of chaos and symmetry in Krause and Robert (KR) flow"],"prefix":"10.1007","volume":"25","author":[{"given":"Anam","family":"Azam","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Muhammad","family":"Aqeel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zakir","family":"Hussain","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,9,24]]},"reference":[{"key":"5325_CR1","doi-asserted-by":"publisher","first-page":"8317","DOI":"10.1007\/s00500-017-2779-0","volume":"22","author":"MA Ahandani","year":"2018","unstructured":"Ahandani MA, Ghiasi AR, Kharrati H (2018) Parameter identification of chaotic systems using a shuffled backtracking search optimization algorithm. Soft Comput 22:8317\u20138339","journal-title":"Soft Comput"},{"key":"5325_CR2","doi-asserted-by":"publisher","first-page":"426","DOI":"10.1140\/epjp\/i2017-11665-7","volume":"132","author":"M Aqeel","year":"2017","unstructured":"Aqeel M, Azam A, Ahmad S (2017) Control of chaos: lie algebraic exact linearization approach for Lu system. Eur Phys J Plus 132:426","journal-title":"Eur Phys J Plus"},{"key":"5325_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11071-017-3641-8","volume":"90","author":"A Azam","year":"2017","unstructured":"Azam A, Aqeel M, Ahmad S, Ahmed F (2017) Chaotic behaviour of modified stretched-twist-fold flow with fractal property. Nonlinear Dyn 90:1\u201312","journal-title":"Nonlinear Dyn"},{"key":"5325_CR4","doi-asserted-by":"publisher","first-page":"3053","DOI":"10.1007\/s11071-018-4544-z","volume":"94","author":"M Balcerzak","year":"2018","unstructured":"Balcerzak M, Pikunov D, Dabrowski A (2018) The fastest, simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems. Nonlinear Dyn 94:3053\u20133065","journal-title":"Nonlinear Dyn"},{"key":"5325_CR5","volume-title":"Stretch, twist, fold: the fast dynamo","author":"S Childress","year":"1995","unstructured":"Childress S, Gilbert AD (1995) Stretch, twist, fold: the fast dynamo. Springer, Berlin"},{"key":"5325_CR6","volume-title":"Symmetry and perturbation theory in nonlinear dynamics Berlin","author":"G Cicogna","year":"1999","unstructured":"Cicogna G, Gaeta G (1999) Symmetry and perturbation theory in nonlinear dynamics Berlin. Springer, New York"},{"key":"5325_CR7","doi-asserted-by":"publisher","first-page":"143","DOI":"10.1016\/0031-9201(86)90087-7","volume":"42","author":"D Crossley","year":"1986","unstructured":"Crossley D, Jensen O (1986) The stochastic excitation of reversals in simple dynamos. Phys Earth Planet Interiors 42:143\u2013153","journal-title":"Phys Earth Planet Interiors"},{"key":"5325_CR8","unstructured":"Gerard L, Slotine J (2006) Neural networks and controlled symmetries, a generic framework"},{"key":"5325_CR9","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780195310658.001.0001","volume-title":"The symmetry of chaos","author":"R Gilmore","year":"2007","unstructured":"Gilmore R, Letellier C (2007) The symmetry of chaos. Oxford University Press, Oxford"},{"key":"5325_CR10","volume-title":"The symmetry perspective: from equilibrium to chaos in phase space and physical space","author":"M Golubitsky","year":"2003","unstructured":"Golubitsky M, Stewart I (2003) The symmetry perspective: from equilibrium to chaos in phase space and physical space. Birkhauser, Berlin"},{"issue":"1","key":"5325_CR11","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/S0034-4877(97)85617-0","volume":"40","author":"WS Koon","year":"1997","unstructured":"Koon WS, Marsden JE (1997) The Hamiltonian and Lagrangian approaches to the dynamics of nonholonomic systems. Rep Math Phys 40(1):21\u201362","journal-title":"Rep Math Phys"},{"key":"5325_CR12","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/0273-1177(81)90344-6","volume":"1","author":"F Krause","year":"1981","unstructured":"Krause F, Roberts PH (1981) Strange attractor character of large-scale nonlinear dynamos. Adv Space Res 1:231\u2013240","journal-title":"Adv Space Res"},{"issue":"2","key":"5325_CR13","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1080\/14029251.2015.1033237","volume":"22","author":"C L\u0103zureanu","year":"2015","unstructured":"L\u0103zureanu C, B\u00eenzar T (2015) Symmetries of some classes of dynamical systems. J Nonlinear Math Phys 22(2):265\u2013274","journal-title":"J Nonlinear Math Phys"},{"key":"5325_CR14","doi-asserted-by":"publisher","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","volume":"20","author":"EN Lorenz","year":"1963","unstructured":"Lorenz EN (1963) Deterministic non periodic flow. J Atmos Sci 20:130","journal-title":"J Atmos Sci"},{"key":"5325_CR15","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2682-6","volume-title":"Introduction to mechanics and symmetry","author":"J Marsden","year":"1994","unstructured":"Marsden J, Ratiu T (1994) Introduction to mechanics and symmetry. Springer, New York"},{"issue":"4","key":"5325_CR16","doi-asserted-by":"publisher","first-page":"041010","DOI":"10.1115\/1.4042030","volume":"141","author":"M Marwan","year":"2019","unstructured":"Marwan M, Ahmad S, Aqeel M, Sabir M (2019) Control analysis of Rucklidge chaotic system. J Dyn Syst Meas Control 141(4):041010","journal-title":"J Dyn Syst Meas Control"},{"key":"5325_CR17","doi-asserted-by":"publisher","first-page":"4333","DOI":"10.1007\/s00500-019-04197-8","volume":"24","author":"M Marwan","year":"2020","unstructured":"Marwan M, Mehboob M, Ahmad S et al (2020) Hopf bifurcation of forced Chen system and its stability via adaptive control with arbitrary parameters. Soft Comput 24:4333\u20134341","journal-title":"Soft Comput"},{"issue":"3","key":"5325_CR18","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1137\/060666044","volume":"6","author":"PG Mehta","year":"2007","unstructured":"Mehta PG, Hagen G, Banaszuk A (2007) Symmetry and symmetry-breaking for a wave equation with feedback. SIAM Dyn Syst Mag 6(3):549\u2013575","journal-title":"SIAM Dyn Syst Mag"},{"key":"5325_CR19","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/0375-9601(93)90735-I","volume":"178","author":"R Miranda","year":"1993","unstructured":"Miranda R, Stone E (1993) The proto-Lorenz system. Phys Lett A 178:105\u2013113","journal-title":"Phys Lett A"},{"key":"5325_CR20","volume-title":"Applications of lie groups to mechanics and symmetry","author":"P Oliver","year":"1986","unstructured":"Oliver P (1986) Applications of lie groups to mechanics and symmetry. Springer, Berlin"},{"key":"5325_CR21","doi-asserted-by":"publisher","first-page":"7469","DOI":"10.1007\/s00500-019-04373-w","volume":"24","author":"K Rajagopal","year":"2019","unstructured":"Rajagopal K, Akgul A, Jafari S et al (2019) An exponential jerk system, its fractional-order form with dynamical analysis and engineering application. Soft Comput 24:7469\u20137479","journal-title":"Soft Comput"},{"issue":"1","key":"5325_CR22","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1017\/S0305004100033223","volume":"54","author":"T Rikitake","year":"1958","unstructured":"Rikitake T (1958) Oscillations of a system of disk dynamos. Proc Camb Philos Soc 54(1):89\u2013105","journal-title":"Proc Camb Philos Soc"},{"key":"5325_CR23","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1016\/0375-9601(76)90101-8","volume":"57","author":"OE Rossler","year":"1976","unstructured":"Rossler OE (1976) An equation for continuous chaos. Phys Lett A 57:397\u2013398","journal-title":"Phys Lett A"},{"issue":"8","key":"5325_CR24","doi-asserted-by":"publisher","first-page":"2507","DOI":"10.1142\/S0218127405013447","volume":"15","author":"C Sarasola","year":"2005","unstructured":"Sarasola C, D\u2019anjou A, Torrealdea FJ, Moujahid A (2005) Energy-like functions for some dissipative chaotic systems. Int J Bifurc Chaos 15(8):2507\u20132521","journal-title":"Int J Bifurc Chaos"},{"issue":"4","key":"5325_CR25","first-page":"3905","volume":"60","author":"EM Shahverdiev","year":"1999","unstructured":"Shahverdiev EM (1999) Boundedness of dynamical systems and chaos synchronization. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip 60(4):3905\u20133909","journal-title":"Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip"},{"issue":"7","key":"5325_CR26","doi-asserted-by":"publisher","first-page":"1025","DOI":"10.1109\/TAC.2005.851449","volume":"50","author":"MW Spong","year":"2005","unstructured":"Spong MW, Bullo F (2005) Controlled symmetries and passive walking. IEEE Trans Autom Control 50(7):1025\u20131031","journal-title":"IEEE Trans Autom Control"},{"key":"5325_CR27","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1016\/0167-2789(85)90011-9","volume":"26","author":"A Wolf","year":"1985","unstructured":"Wolf A, Swift JB, Swinney HL, Wastano JA (1985) Determining Lyapunov exponents from time series. Physica D 26:285\u2013317","journal-title":"Physica D"},{"issue":"8","key":"5325_CR28","doi-asserted-by":"publisher","first-page":"080510","DOI":"10.1088\/1674-1056\/19\/8\/080510","volume":"19","author":"J Ying","year":"2010","unstructured":"Ying J, Qin-Sheng B (2010) Symmetric bursting behaviour in non-smooth Chua\u2019s circuit. Chin Phys B 19(8):080510","journal-title":"Chin Phys B"}],"container-title":["Soft Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05325-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00500-020-05325-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00500-020-05325-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,14]],"date-time":"2024-08-14T19:01:10Z","timestamp":1723662070000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00500-020-05325-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,24]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2021,2]]}},"alternative-id":["5325"],"URL":"https:\/\/doi.org\/10.1007\/s00500-020-05325-5","relation":{},"ISSN":["1432-7643","1433-7479"],"issn-type":[{"type":"print","value":"1432-7643"},{"type":"electronic","value":"1433-7479"}],"subject":[],"published":{"date-parts":[[2020,9,24]]},"assertion":[{"value":"24 September 2020","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Compliance with ethical standards"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}