{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,8]],"date-time":"2026-07-08T16:50:10Z","timestamp":1783529410479,"version":"3.55.0"},"reference-count":61,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T00:00:00Z","timestamp":1765497600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"crossref","award":["417698841"],"award-info":[{"award-number":["417698841"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>Bounds for positive definite sets such as attractors of dynamic systems are typically characterized by Lyapunov-like functions. These Lyapunov functions and their time derivatives must satisfy certain definiteness conditions, whose verification usually requires considerable experience. If the system and a Lyapunov-like candidate function are polynomial, the definiteness conditions lead to Boolean combinations of polynomial equations and inequalities with quantifiers that can be formally solved using quantifier elimination. Unfortunately, the known algorithms for quantifier elimination require considerable computing power, meaning that many problems cannot be solved within a reasonable amount of time. In this context, it is particularly important to find a suitable mathematical formulation of the problem. This article develops a method that reduces the expected computational effort required for the necessary verification of definiteness conditions. The approach is illustrated using the example of the Chua system with cubic nonlinearity.<\/jats:p>","DOI":"10.3390\/a18120785","type":"journal-article","created":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T11:13:33Z","timestamp":1765538013000},"page":"785","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Computation of Bounds for Polynomial Dynamic Systems"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3347-0864","authenticated-orcid":false,"given":"Klaus","family":"R\u00f6benack","sequence":"first","affiliation":[{"name":"Institute of Control Theory, Faculty of Electrical and Computer Engineering, TUD Dresden University of Technology, 01062 Dresden, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8527-8727","authenticated-orcid":false,"given":"Daniel","family":"Gerbet","sequence":"additional","affiliation":[{"name":"Institute of Control Theory, Faculty of Electrical and Computer Engineering, TUD Dresden University of Technology, 01062 Dresden, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,12]]},"reference":[{"key":"ref_1","unstructured":"Slotine, J.J.E., and Li, W. (1991). Applied Nonlinear Control, Prentice-Hall."},{"key":"ref_2","unstructured":"Khalil, H.K. (2015). Nonlinear Control, Pearson."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1002\/zamm.19870671215","article-title":"Attraktorlokalisierung des Lorenz-Systems","volume":"67","author":"Leonov","year":"1987","journal-title":"ZAMM-Appl. Math. Mech."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Sparrow, C. (1982). The Lorenz Equations: Birfucations, Chaos, and Strange Atractors, Springer.","DOI":"10.1007\/978-1-4612-5767-7"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1016\/j.chaos.2004.05.021","article-title":"Estimating the bounds for the Lorenz family of chaotic systems","volume":"23","author":"Li","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_6","unstructured":"Reitmann, V., and Leonov, G.A. (1987). Attraktoreingrenzung f\u00fcr nichtlineare Systeme. Teubner-Texte zur Mathematik, BSB Teubner."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Kuznetsov, N., and Reitmann, V. (2020). Attractor Dimension Estimates for Dynamical Systems: Theory and Computation, Springer.","DOI":"10.1007\/978-3-030-50987-3"},{"key":"ref_8","unstructured":"Tarski, A. (1948). A Decision Method for Elementary Algebra and Geometry, Rand Corporation. Project rand."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Caviness, B.F., and Johnson, J.R. (1998). Quantifier Elimination and Cylindical Algebraic Decomposition, Springer.","DOI":"10.1007\/978-3-7091-9459-1"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1016\/j.nahs.2009.04.010","article-title":"A semi-algebraic approach for asymptotic stability analysis","volume":"3","author":"She","year":"2009","journal-title":"Nonlinear Anal. Hybrid Syst."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Tong, J., and Bajcinca, N. (2018, January 12\u201315). Computation of Feasible Parametric Regions for Common Quadratic Lyapunov Functions. Proceedings of the European Control Conference (ECC), Limassol, Cyprus.","DOI":"10.23919\/ECC.2018.8550205"},{"key":"ref_12","first-page":"25","article-title":"Automatic Generation of Bounds for Polynomial Systems with Application to the Lorenz System","volume":"113C","author":"Richter","year":"2018","journal-title":"Chaos Solitons Fractals"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"074502","DOI":"10.1115\/1.4043380","article-title":"Calculating Positive Invariant Sets: A Quantifier Elimination Approach","volume":"14","author":"Richter","year":"2019","journal-title":"J. Comput. Nonlinear Dyn."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"e202000161","DOI":"10.1002\/pamm.202000161","article-title":"Formal Calculation of Positive Invariant Sets for the Lorenz Family Combining Lyapunov Approaches and Quantifier Elimination","volume":"20","year":"2021","journal-title":"Proc. Appl. Math. Mech."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"e202200197","DOI":"10.1002\/pamm.202200197","article-title":"On the Systematic Construction of Lyapunov Functions for Polynomial Systems","volume":"23","author":"Natkowski","year":"2023","journal-title":"Proc. Appl. Math. Mech."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1597","DOI":"10.1088\/0951-7715\/16\/5\/303","article-title":"An ultimate bound on the trajectories of the Lorenz system and its applications","volume":"16","author":"Pogromsky","year":"2003","journal-title":"Nonlinearity"},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Isidori, A. (1999). Nonlinear Control Systems II, Springer.","DOI":"10.1007\/978-1-4471-0549-7"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1669","DOI":"10.1007\/s10625-006-0003-6","article-title":"Localization of invariant compact sets of dynamical systems","volume":"41","author":"Krishchenko","year":"2005","journal-title":"Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1080\/00029890.1991.11995702","article-title":"Positive definite matrices and Sylvester\u2019s criterion","volume":"98","author":"Gilbert","year":"1991","journal-title":"Am. Math. Mon."},{"key":"ref_20","unstructured":"Marcus, M., and Minc, H. (1992). A Survey of Matrix Theory and Matrix Inequalities, Dover."},{"key":"ref_21","unstructured":"Tibken, B., and Dilaver, K.F. (2002, January 10\u201313). Computation of subsets of the domain of attraction for polynomial systems. Proceedings of the 41st IEEE Conference on Decision and Control, Las Vegas, NV, USA."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"218","DOI":"10.9746\/jcmsi.5.218","article-title":"Sum of Squares Based Input-to-State Stability Analysis of Polynomial Nonlinear Systems","volume":"5","author":"Ichihara","year":"2012","journal-title":"SICE J. Control. Meas. Syst. Integr."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"365","DOI":"10.2307\/1969640","article-title":"A New Decision Method for Elementary Algebra","volume":"60","author":"Seidenberg","year":"1954","journal-title":"Ann. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Collins, G.E. (1975, January 20\u201323). Quantifier elimination for real closed fields by cylindrical algebraic decompostion. Proceedings of the Automata Theory and Formal Languages 2nd GI Conference Kaiserslautern, Kaiserslautern, Germany.","DOI":"10.1007\/3-540-07407-4_17"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0747-7171(08)80152-6","article-title":"Partial Cylindrical Algebraic Decomposition for Quantifier Elimination","volume":"12","author":"Collins","year":"1991","journal-title":"J. Symb. Comput."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1016\/S0747-7171(88)80003-8","article-title":"The complexity of linear problems in fields","volume":"5","author":"Weispfenning","year":"1988","journal-title":"J. Symb. Comput."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Sturm, T. (2018). Thirty Years of Virtual Substitution: Foundations, Techniques, Applications. Proceedings of the 2018 ACM on International Symposium on Symbolic and Algebraic Computation, ACM.","DOI":"10.1145\/3208976.3209030"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Gonzalez-Vega, L., Lombardi, H., Recio, T., and Roy, M.F. (1989, January 17\u201319). Sturm-Habicht Sequence. Proceedings of the ACM-SIGSAM 1989 International Symposium on Symbolic and Algebraic Computation, Portland, OR, USA.","DOI":"10.1145\/74540.74558"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0747-7171(88)80004-X","article-title":"Real quantifier elimination is doubly exponential","volume":"5","author":"Davenport","year":"1988","journal-title":"J. Symb. Comput."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Brown, C.W., and Davenport, J.H. (August, January 28). The complexity of quantifier elimination and cylindrical algebraic decomposition. Proceedings of the 2007 International Symposium on Symbolic and Algebraic Computation, Waterloo, ON, Canada.","DOI":"10.1145\/1277548.1277557"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Weispfenning, V. (1994, January 20\u201322). Quantifier Elimination for Real Algebra \u2013 the Cubic Case. Proceedings of the International Symposium on Symbolic and Algebraic Computation (ISSAC), Oxford, UK.","DOI":"10.1145\/190347.190425"},{"key":"ref_32","unstructured":"Ko\u0161ta, M., and Sturm, T. (2015). A Generalized Framework for Virtual Substitution. arXiv."},{"key":"ref_33","unstructured":"Ko\u0161ta, M. (2016). New Concepts for Real Quantifier Elimination by Virtual Substitution. [Ph.D. Thesis, Universit\u00e4t des Saarlandes, Fakult\u00e4t f\u00fcr Mathematik und Informatik]."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1145\/968708.968710","article-title":"QEPCAD B: A program for computing with semi-algebraic sets using CADs","volume":"37","author":"Brown","year":"2003","journal-title":"ACM SIGSAM Bull."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"2","DOI":"10.1145\/261320.261324","article-title":"Redlog: Computer algebra meets computer logic","volume":"31","author":"Dolzmann","year":"1997","journal-title":"ACM SIGSAM Bull."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/11870814_7","article-title":"Efficient preprocessing methods for quantifier elimination","volume":"4194","author":"Brown","year":"2006","journal-title":"Proceedings of the CASC. Lecture Notes in Computer Science"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"518","DOI":"10.1007\/978-3-662-44199-2_78","article-title":"SyNRAC: A Toolbox for Solving Real Algebraic Constraints","volume":"8592","author":"Hong","year":"2014","journal-title":"Proceedings of the Mathematical Software\u2014ICMS 2014, Lecture Notes in Computer Science"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.jsc.2015.11.008","article-title":"Quantifier elimination by cylindrical algebraic decomposition based on regular chains","volume":"75","author":"Chen","year":"2016","journal-title":"J. Symb. Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"130","DOI":"10.1175\/1520-0469(1963)020<0130:DNF>2.0.CO;2","article-title":"Deterministic non-periodic flow","volume":"20","author":"Lorenz","year":"1963","journal-title":"J. Atmos. Sci."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s11071-010-9702-x","article-title":"Yang and Yin parameters in the Lorenz system","volume":"62","author":"Ge","year":"2010","journal-title":"Nonlinear Dyn."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"328","DOI":"10.1016\/j.cnsns.2015.06.034","article-title":"Takens\u2013Bogdanov bifurcations of equilibria and periodic orbits in the Lorenz system","volume":"30","author":"Algaba","year":"2016","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/j.physleta.2005.12.104","article-title":"Localization of compact invariant sets of the Lorenz system","volume":"353","author":"Krishchenko","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"844","DOI":"10.1016\/j.jmaa.2005.11.008","article-title":"Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system","volume":"323","author":"Li","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1016\/j.jmaa.2005.12.034","article-title":"Geometric existence theory for the control-affine H\u221e problem","volume":"324","author":"McCaffrey","year":"2006","journal-title":"J. Math. Anal. Appl."},{"key":"ref_45","doi-asserted-by":"crossref","unstructured":"Abou-Kandil, H., Freiling, G., Ionescu, V., and Jank, G. (2003). Symmetric differential Riccati equations: An operator based approach. Matrix Riccati Equations in Control and Systems Theory, Springer.","DOI":"10.1007\/978-3-0348-8081-7"},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1109\/TCS.1984.1085459","article-title":"A chaotic attractor from Chua\u2019s circuit","volume":"31","author":"Matsumoto","year":"1984","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_47","first-page":"250","article-title":"The genesis of Chua\u2019s circuit","volume":"46","author":"Chua","year":"1992","journal-title":"Arch. Elektron. \u00dcbertragungstechnik (AE\u00dc)"},{"key":"ref_48","first-page":"66","article-title":"Robust OP AMP realization of Chua\u2019s circuit","volume":"3\u20134","author":"Kennedy","year":"1992","journal-title":"Frequenz"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"934","DOI":"10.1109\/81.340866","article-title":"Implementation of Chua\u2019s Circuit with a Cubic Nonlinearity","volume":"41","author":"Zhong","year":"1994","journal-title":"IEEE Trans. Circuits Syst. I"},{"key":"ref_50","unstructured":"R\u00f6benack, K. (2005). Regler- und Beobachterentwurf f\u00fcr nichtlineare Systeme mit Hilfe des Automatischen Differenzierens, Shaker Verlag."},{"key":"ref_51","unstructured":"The Sage Developers. SageMath, the Sage Mathematics Software System, Version 10.2. Available online: https:\/\/www.sagemath.org."},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"376","DOI":"10.1111\/j.1749-6632.1979.tb29482.x","article-title":"Continuous Chaos \u2014 Four Prototyp Equations","volume":"316","year":"1979","journal-title":"Ann. N. Y. Acad. Sci."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"R647","DOI":"10.1103\/PhysRevE.50.R647","article-title":"Some simple chaotic flows","volume":"50","author":"Sprott","year":"1994","journal-title":"Phys. Rev. E"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"758","DOI":"10.1119\/1.19538","article-title":"Simple chaotic systems and circuits","volume":"68","author":"Sprott","year":"2000","journal-title":"Am. J. Phys."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"1330002","DOI":"10.1142\/S0218127413300024","article-title":"Hidden attractors in dynamical systems. From hidden oscillations in Hilbert\u2013Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits","volume":"23","author":"Leonov","year":"2013","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"5445","DOI":"10.3182\/20140824-6-ZA-1003.02501","article-title":"Hidden attractors in dynamical systems: Systems with no equilibria, multistability and coexisting attractors","volume":"47","author":"Kuznetsov","year":"2014","journal-title":"IFAC Proc. Vol."},{"key":"ref_57","doi-asserted-by":"crossref","unstructured":"Sepulchre, R., Jankovi\u0107, M., and Kokotovi\u0107, P. (1997). Constructive Nonlinear Control, Springer.","DOI":"10.1007\/978-1-4471-0967-9"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1002\/oca.2733","article-title":"Synthesis of control Lyapunov functions and stabilizing feedback strategies using exit-time optimal control Part II: Numerical approach","volume":"42","author":"Yegorov","year":"2021","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"1507","DOI":"10.1016\/j.physd.2009.03.002","article-title":"Almost-invariant sets and invariant manifolds \u2013 Connecting probabilistic and geometric descriptions of coherent structures in flows","volume":"238","author":"Froyland","year":"2009","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.ifacol.2023.02.004","article-title":"Examples for separable control Lyapunov functions and their neural network approximation","volume":"56","author":"Sperl","year":"2023","journal-title":"IFAC-PapersOnLine"},{"key":"ref_61","unstructured":"(2025, November 01). Computation of Bounds for Polynomial Dynamic Systems, Source Files. Available online: https:\/\/github.com\/TUD-RST\/computation-bounds-polynomial-systems."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/785\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,14]],"date-time":"2025-12-14T05:20:32Z","timestamp":1765689632000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/12\/785"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,12]]},"references-count":61,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["a18120785"],"URL":"https:\/\/doi.org\/10.3390\/a18120785","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,12]]}}}