Nonlinear chemical kinetic schemes derived from mechanical and electrical dynamical systems

The nonlinear transformation presented converts a mechanical or electrical system with positive and negative variables into a dynamically related chemical scheme that allows only nonnegative state variables. This transformation preserves the phase space qualitative features of the original system. The harmonic oscillator, Van der Pol, Lorenz, Rossler spiral chaos, forced negative stiffness Duffing, and a series RLC‐circuit are transformed into mass action chemical schemes. For example, the harmonic oscillator is converted into the Lotka–Volterra model. Hence, the information from a variety of physical studies is applicable to chemical systems.

[1]  Alfred Brian Pippard,et al.  The physics of vibration , 1991 .

[2]  R. May,et al.  Nonlinear Aspects of Competition Between Three Species , 1975 .

[3]  R. J. Field,et al.  Oscillations and Traveling Waves in Chemical Systems , 1985 .

[4]  Nikola Samardzija,et al.  Stability properties of autonomous homogeneous polynomial differential systems , 1983 .

[5]  Armin Schmidt,et al.  Zur Umsetzung von Trichloracetimidsäuremethylester mit Antimon(V)-chlorid. , 1976 .

[6]  Y. Pomeau,et al.  Order within chaos , 1986 .

[7]  H. Pain,et al.  The physics of vibrations and waves , 1968 .

[8]  C. Sparrow.  The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors , 1982 .

[9]  Francis C. Moon.  Fractal Boundary for Chaos in a Two-State Mechanical Oscillator , 1984 .

[10]  Bruce L. Clarke,et al.  Complete set of steady states for the general stoichiometric dynamical system , 1981 .

[11]  P. Holmes,et al.  A nonlinear oscillator with a strange attractor , 1979, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[12]  Françoise Argoul,et al.  Chemical Chaos: From Hints to Confirmation , 1987 .

[13]  J. Tyson.  Some further studies of nonlinear oscillations in chemical systems , 1973 .

[14]  H. Davis.  Introduction to Nonlinear Differential and Integral Equations , 1964 .

[15]  J. N. Holmes.  Speech communication with computers , 1979 .

[16]  P. Cvitanović.  Universality in Chaos , 1989 .

[17]  J. Higgins.  The Theory of Oscillating Reactions - Kinetics Symposium , 1967 .

[18]  S. E. Khaikin,et al.  Theory of Oscillators , 1966 .

[19]  I. Epstein.  Oscillations and chaos in chemical systems , 1983 .

[20]  Li,et al.  Fractal basin boundaries and homoclinic orbits for periodic motion in a two-well potential. , 1985, Physical review letters.

[21]  S. Smale.  On the differential equations of species in competition , 1976, Journal of mathematical biology.

[22]  O. Rössler.  An equation for continuous chaos , 1976 .

[23]  John J. Tyson,et al.  Properties of two‐component bimolecular and trimolecular chemical reaction systems , 1973 .

[24]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[25]  C. Truesdell,et al.  Essays in the History of Mechanics , 1968 .

[26]  A. J. Lotka.  UNDAMPED OSCILLATIONS DERIVED FROM THE LAW OF MASS ACTION. , 1920 .

[27]  V. Hlavácek,et al.  Dynamics of nonlinear systems , 1986 .

[28]  Katsuhiko Ogata,et al.  Modern Control Engineering , 1970 .

[29]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .