Singularities of transition processes in dynamical systems: Qualitative theory of critical delays
DOI:
https://doi.org/10.58997/ejde.mon.05Keywords:
Dynamical system; transition process; relaxation time; bifurcation; limit set; Smale order.Abstract
This monograph presents a systematic analysis of the singularities in the transition processes for dynamical systems. We study general dynamical systems, with dependence on a parameter, and construct relaxation times that depend on three variables: Initial conditions, parameters \(k\) of the system, and accuracy \(\varepsilon\) of the relaxation. We study the singularities of relaxation times as functions of \((x_0,k)\) under fixed \(\varepsilon\), and then classify the bifurcations (explosions) of limit sets. We study the relationship between singularities of relaxation times and bifurcations of limit sets. An analogue of the Smale order for general dynamical systems under perturbations is constructed. It is shown that the perturbations simplify the situation: the interrelations between the singularities of relaxation times and other peculiarities of dynamics for general dynamical system under small perturbations are the same as for the Morse-Smale systems.
For more information see https://ejde.math.txstate.edu/Monographs/05/abstr.html
References
Afraimovich V.S., Bykov V.V., Shilnikov L.P., Rise and structure of Lorenz attractor, Dokl. AN SSSR 234, 2 (1977), 336–339.
Aleksandrov P.S., Passynkov V.A. Introduction into dimension theory. Moscow: Nauka, 1973. 574 p.
Andronov A.A., Leontovich E.A., Gordon I.I., Mayer A.G. Bifurcations theory for dynamical systems on the plane. Moscow: Nauka, 1967. 488 p.
Andronov A.A., Vitt A.A., Khaikin S.E. Oscillation theory. Moscow: Nauka, 1981. 568 p.
Anosov D.V., Geodesic flows on closed Riemannian manifolds of negative curvature, Proc. V.A. Steklov Math. Inst. 90 (1967).
Anosov D.V., About one class of invariant sets of smooth dynamical systems, Proceedings of International conference on non-linear oscillation, V.2. Kiev, 1970, 39–45.
Auslander J., Filter stability in dynamic systems, SIAM J. Math. Anal., 8, 3 (1977), 573–579.
de Baggis G.F., Rough systems of two differential equations, Uspekhi Matematicheskikh Nauk 10, 4 (1955), 101–126.
Barbashin E.A., Introduction into stability theory. Moscow, Nauka, 1967. 223 p.
Bykov V.I., Gorban A.N., Pushkareva T.P., Singularities of relaxation times in oxidation reaction of CO on Pt, Teor. i exp. khimija 18, 4 (1982), 431–439.
Arnold V.I., Small denominators and stability problems in classical and celestial mechanics, Uspekhi Matematicheskikh Nauk 18, 6 (1963), 91–192.
Arnold V.I. Additional chapters of the theory of ordinary differential equations. Moscow: Nauka, 1978. 304 p.
Arnold V.I., Varchenko A.N., Gusein-Zade S.M. The singularity of differentiable mappings. V.1, Moscow: Nauka, 1982; V.2, Moscow: Nauka, 1984.
Bajaj P.N., Stability in semiflows, Math. Mod. Econom., Amsterdam e.a., 1974, 403–408.
Bautin N.N., Leontovich E.A. Methods and ways of qualitative exploration of dynamical systems on the plane. Moscow: Nauka, 1976. 496 p.
Birkhof J.D., Dynamical systems. Moscow– Leningrad: Gostekhizdat, 1941. 320 p.
Bogolubov N.N., Selected works on statistical physics. Moscow: Moscow State University Press, 1979, p.193–269.
Bowen R., Methods of symbolical dynamics. Moscow: Mir, 1979. 245 p.
Bourbaki N., General topology. General structures. Moscow: Nauka, 1968. 272 p.
Das K.M., Lakshmi S.S., Negative and weak negative attractors in semidynamical systems, Func. ekvacioj 21, 1 (1978), 53–62.
Dobrynskii V.A., Sharkovskii A.N., Typicalness of dynamical systems, whose almost all trajectories are stable under uniformly influencing perturbations, Dokl. AN SSSR 211, 2 (1973), 273–276.
Duboshin G.N., On the problem of motion stability under uniformly influencing perturbations, Proceedings P.K. Sternberg Inst. of Astronomy 14, 1 (1940), 153–164.
Easton H., Chain transitivity and the domain of influence of an invariant set, Lect. Notes Math., V.668 (1978), p.95–102.
Eigen M., De Maeyer Leo., Theoretical basis of relaxation spectroscopy, In: Investigations of rates and mechanisms of reactions. Ed by G.G.Hammes. Jon Willey & Sons, 1974.
Elokhin V.I., Bykov V.I., Slinko M.G., Yablonskii G.S., Some problems of dynamics of oxidation reaction of CO on Pt, Dokl. AN SSSR 238, 3 (1978), 615–618.
Franke J.E., Selgrade J.F., Hyperbolicity and chain recurrence, J. Different. Equat. 26, 1 (1977), 27–36.
Franke J.E., Selgrade J.F., Hyperbolicity and cycles, Trans. Amer. Math. Soc. 245 (1978), 251–262.
Germaidze V.E., Krasovskii N.N., On the stability under uniformly influencing perturbations, Prikl. matem. i mech. 21, 6 (1957), 769–775.
Glazman I.M., Relaxation on the surface with saddle points, Dokl. AN SSSR 161, 4 (1965), 750–752.
Gorban A.N., Slow relaxations of perturbed systems. I (general results). Krasnoyarsk: Preprint Computing Center USSR Academy of Sciences, 1980.
Gorban A.N., Slow relaxations and bifurcations of omega-limit sets of dynamical systems,
PhD Thesis in Physics & Math (Differential Equations & Math.Phys), Kuibyshev, Russia, 1980.
Gorban A.N., Cheresiz V.M., Slow relaxations and bifurcations of limit sets of dynamical systems. Krasnoyarsk: Preprint Computing Center USSR Academy of Sciences, 1980.
Gorban A.N., Cheresiz V.M., Slow relaxations of dynamical systems and bifurcations of limit sets, Dokl. AN SSSR 261, 5 (1981), 1050–1054.
Gorban A.N., Cheresiz V.M., Elokhin V.I., Yablonskii G.S., Slow relaxations and critical phenomena in catalytic reactions, In: Mathematical methods in chemistry. V.2. Qualitative methods. Moscow: ZNIITEneftehim, 1980, 53–60.
Gorban A.N., Elokhin V.I., Cheresiz V.M., Yablonskii G.S., Possible kinetic reasons of slow relaxations of catalytic reactions, In: Non-stationary processes in catalysis. Part 1. Novosibirsk, 1979, 83–88.
Gottschalk W.H., Bibliography for topological dynamics, Wesleyan Univ., 1966, 76 p.
Gottschalk W.H., Hedlund G.A., Topological dynamics, Providence, Amer. Math. Soc. Colloquium Publ. 36, 1 (1955), 1–51.
Gromov M., Entropy, homology and semialgebraic geometry. S´eminaire Bourbaki, Vol. 1985/86. AstSerisque 145-146, (1987), 225-240.
Hasselblatt, B., Katok, A. (Eds.), Handbook of Dynamical Systems, Vol. 1A, Elsevier, 2002.
Kaloshin V.Yu. Prevalence in spaces of smooth mappings and a prevalent Kupka-Smale theorem, Dokl. Akad. Nauk 355, 3 (1997), 306–307.
Katok A.B., Local properties of hyperbolic sets. Supplement I to [57], 214–232.
Katok A., Hasselblat, B. Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math. and its Applications, Vol. 54, Cambridge University Press, 1995.
Kifer Yu.I., Some results concerning small stochastic perturbations of dynamical systems, Teoriya verojatnostej i ee primenenija 19, 2 (1974), 514–532.
Kifer Yu.I., About small stochastic perturbations of some smooth dynamical systems, Izv. AN SSSR. Matematika 38, 5 (1974), 1091–1115.
Kornfeld I.P., Sinai Ya.G., Fomin S.V., Ergodic theory, Moscow: Nauka, 1980. 383 p.
Krasovskii N.N., Some problems of the motion stability theory, Moscow: Fizmatgiz, 1959.
Kupka I., Contribution `a la th´eorie des champs g´en´eriques. Contrib. Diff. Eqs. 2 (1963), 457–484. Ibid 3, (1964), 411–420.
Kuratovskii K., Topology, V.1 Moscow: Mir, 1966. 594 p.
Kuratovskii K., Topology. V.2 Moscow: Mir, 1969. 624 p.
Lorenz E.N., Deterministic nonperiodic flow, J. Atmos. Sci. 20, 1 (1963), 130–141.
Malkin I.G., On the stability under uniformly influencing perturbations, Prikl. matem. imech. 8, 3 (1944), 241–245.
Malkin I.G., The motion stability theory, Moscow: Nauka, 1966, 20–21 and 301–305.
Martynjuk A.A., Gutovsky R., Integral inequalities and motion stability, Kiev: Naukova dumka, 1979, p.139–179.
McCan C., Negative escape time in semidynamical systems, Func. ekvacioj 20, 1 (1977), 37–39.
Moser J., Convergent series expansions for quasiperiodic motions, Math. Ann. 169 (1977), 136–176.
Nemytskii V.V., Stepanov V.V., Qualitative theory of differential equations, Moscow–Leningrad.: Gostekhizdat, 1949. 550 p.
Nitetsky Z., Introduction into differential dynamics, Moscow: Mir, 1975. 305 p.
Piexoto M., Structural stability on two–dimensional manifolds, Topology 1, 1 (1962), 101–120.
Pontrjagin L.S., Andronov A.A., Vitt A.A., On statistical consideration of dynamical systems, Sov. Phys. JETPh 3, 3 (1933), 165–180.
Poston T., Stuart I., Catastrophe theory and its applications. Moscow: Mir, 1980.
Pugh Ch., The closing lemma, Amer. J. Math. 89, 4 (1967), 956–1009.
Sharkovskii A.N., Stability of trajectories and dynamical systems in whole, In: 9-th Summer Mathematical School, Kiev: Naukova dumka, 1976, p.349–360.
Sharkovskii A.N., Structural theory of differentiable dynamical systems and weakly nonwandering points, Abh. Akad. Wiss. DDR. Abt. Math., Naturwiss., Techn. No.4 (1977), 193–200.
Semjonov N.N., Chain reactions, Leningrad: Goskhimizdat, 1934. 555 p.
Sibirskii K.S., Introduction into topological dynamics, Kishinev: AS Mold. SSR Press, 1970. 144 p.
Smale S., Stable manifolds for differential equations and diffeomorphisms, Ann. Sc. Norm. Sup. Pisa 18, (1963), 97–116.
Smale S., Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491–496.
Smale S., Differentiable dynamical systems, Bull. AMS 73 (1967), 747–817.
Spivak S.I., Shabat A.B., Shmelev A.S., On induction period of chemical reactions, In: Nonstationary processes in catalysis. Part 1. Novosibirsk: Inst. of Catalysis, 1979, 118–121.
Stepanjanz G.A., On general propositions of qualitative theory of discrete and irreversible continuous systems, Problemy Kibernetiki 44, (1979), 5–14.
Strauss A., Yorke A.J., Identifying perturbations which preserved asymptotic stability, Bull. Amer. Math. Soc. 22, 2 (1969), 513–518.
Trezepizur A., Ombach J., Limit sets on pseudo-dynamical systems, Bull. Acad. Pol. Sci. Ser. Math. Sci., Astron. et Phys. 22, 2 (1974), 157–160.
Van’t-Hoff J.H., Studies in dynamic chemistry. Leningrad.: ONTI, 1936. 178 p.
Ventzel A.D., Freidlin M.I., Small stochastic perturbations of dynamical systems, Uspekhi Matematicheskikh Nauk 25, 1 (1970), 3–55.
Ventzel A.D., Freidlin M.I., Some problems concerning stability under small stochastic perturbations, Teoriya verojatnostej i ee primenenija 17, 2 (1972), 281–295.
Ventzel A.D., Freidlin M.I., Fluctuations in dynamical systems influenced by small stochastic perturbations, Moscow: Nauka, 1979. 424 p.
Walters P., On the pseudoorbit tracing property and its relationship to stability, Lect. Notes Math. V. 668 (1978), 231–244.
Yablonskii G.S., Bykov V.I., Gorban A.N., Elokhin V.I., Kinetic models of catalytic reactions, The series of monographs “Comprehensive Chemical Kinetics”, Vol. 32, Compton R.G. ed., Elsevier, Amsterdam, 1991. XIV+396 p.
Yomdin Y., A quantitative version of the Kupka–Smale theorem, Ergodic Theory Dynamical Systems 5, 3 (1985), 449-472.
Zubov V.I., Stability of motion. Moscow: Vysshaya shkola, 1973. 272 p.
