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Applications of dynamical systems with symmetry

By Peter Brian Ashwin


This thesis examines the application of symmetric dynamical systems theory to\ud two areas in applied mathematics: weakly coupled oscillators with symmetry, and\ud bifurcations in flame front equations.\ud After a general introduction in the first chapter, chapter 2 develops a theoretical\ud framework for the study of identical oscillators with arbitrary symmetry group under an\ud assumption of weak coupling. It focusses on networks with 'all to all' Sn coupling. The\ud structure imposed by the symmetry on the phase space for weakly coupled oscillators\ud with Sn, Zn or Dn symmetries is discussed, and the interaction of internal symmetries\ud and network symmetries is shown to cause decoupling under certain conditions.\ud Chapter 3 discusses what this implies for generic dynamical behaviour of coupled\ud oscillator systems, and concentrates on application to small numbers of oscillators (three\ud or four). We find strong restrictions on bifurcations, and structurally stable heteroclinic\ud cycles.\ud Following this, chapter 4 reports on experimental results from electronic oscillator\ud systems and relates it to results in chapter 3. In a forced oscillator system, breakdown\ud of regular motion is observed to occur through break up of tori followed by a symmetric\ud bifurcation of chaotic attractors to fully symmetric chaos.\ud Chapter 5 discusses reduction of a system of identical coupled oscillators to phase\ud equations in a weakly coupled limit, considering them as weakly dissipative Hamiltonian\ud oscillators with very weakly coupling. This provides a derivation of example phase\ud equations discussed in chapter 2. Applications are shown for two van der Pol-Duffing\ud oscillators in the case of a twin-well potential.\ud Finally, we turn our attention to the Kuramoto-Sivashinsky equation. Chapter 6\ud starts by discussing flame front equations in general, and non-linear models in particular.\ud The Kuramoto-Sivashinsky equation on a rectangular domain with simple\ud boundary conditions is found to be an example of a large class of systems whose linear\ud behaviour gives rise to arbitrarily high order mode interactions.\ud Chapter 7 presents computation of some of these mode interactions using competerised\ud Liapunov-Schmidt reduction onto the kernel of the linearisation, and investigates\ud the bifurcation diagrams in two parameters

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  1. (1980). A classical introduction to modern number theory. Graduate texts in maths. doi
  2. A nonlinear wave equation in nonadiabatic Same propagation. doi
  3. A perturbation method for nonlinear dispersive wave problems. doi
  4. An analytical and numerical study of the bifurcations in a system of linearly coupled oscillators. doi
  5. An asymptotic derivation of two models in flame theory associated with the constant density approximation. doi
  6. (1985). An introduction to the finite element method with applications to nonlinear problems. Wiley-Interscience,
  7. (1956). An introduction to the theory of numbers. doi
  8. (1981). Applications of centre manifold theory. doi
  9. (1987). Asymptotic methods for relaxation oscillations and applications, doi
  10. (1961). Asymptotic methods in nonlinear oscillations. Hindustan Pub.
  11. (1988). Attractor crowding in oscillator arrays. doi
  12. (1985). Attractors representing turbulent flows, doi
  13. (1985). Averaging methods in nonlinear dynamical systems. doi
  14. (1989). Bifurcation and pattern formation in combustion. doi
  15. Biological -rhythms and the behaviour of populations of coupled oscillators. doi
  16. (1989). Boundaxy conditions as symmetry constraints. doi
  17. (1991). Chaos and nonisochronism in weakly coupled oscillators. doi
  18. (1984). Chemical oscillations, waves and turbulence, doi
  19. (1984). Collective dynamics of coupled oscillators with random pinning. doi
  20. (1971). Combustion driven oscillations in industry. doi
  21. (1985). Combustion Theory. Benjamin Cummings,
  22. Convection in a rotating layer: a simple case of turbulence. doi
  23. (1989). Coupled arrays of Josephson junctions and bifurcations of maps with S(n) symmetry. doi
  24. (1990). Coupled nonlinear oscillators and the symmetries of animal gaits. doi
  25. Coupled stationary bifurcations in non-flux boundary value problems. doi
  26. Crises, sudden changes in chaotic attractors and transient chaos. doi
  27. Diffusion-induced chaos in reaction systems. doi
  28. (1989). Dynamics and analysis of patterns. In
  29. (1989). Dynamics of Josephson junction arrays. doi
  30. Eigenvalues of the Laplacian in two dimensions. doi
  31. (1939). Elementary number theory. doi
  32. (1991). Encyclopaedia of mathematical sciences. doi
  33. (1985). Flame propagation in channels: secondary bifurcation to quasi-periodic pulsation. doi
  34. Forced oscillations in a circuit with nonlinear resistance. doi
  35. (1983). Geometrical methods in the theory of ODEs, doi
  36. (1988). Global bifurcations and chaos, doi
  37. (1988). Global bifurcations in flows. In doi
  38. (1988). Global bifurcations of periodic solutions with symmetry, volume 1309 of Springer lecture notes in mathematics. doi
  39. (1979). Group theoretic methods in bifurcation theory, doi
  40. (1988). Groups and singularities in bifurcation theory doi
  41. (1965). Handbook of Mathematical Functions. New Yorl-, doi
  42. (1985). Hopf bifurcation in the presence of symmetry. doi
  43. Hopf bifurcation with the symmetry of the square. doi
  44. (1990). Identical oscillators with symmetry. doi
  45. (1982). Instability and dynamic pattern formation in cellular networks. doi
  46. Interaction of pulsating and spinning waves in nonadiabatic flame propagation. doi
  47. (1977). InvaHant Manifolds,
  48. Isochrons and phaseless sets. doi
  49. Kuramoto-Sivashinsky dynamics on the center-unstable manifold. doi
  50. (1984). Liapounov exponents for the Kuramoto-Sivashisky model. doi
  51. Mode analysis of a ring of a large number of mutually coupled van der Pol oscillators. doi
  52. Mutually synchronised relaxation oscillators as prototypes of oscillating systems in biology. doi
  53. (1985). Nilpotent normal forms and representation theory of SL(2, R). doi
  54. Nonlinear analysis of hydrodynamic instability in flames- I. derivation of the basic equations. doi
  55. (1977). Nonlinear analysis of hydrodynamic instability in flames- II. numerical experiments. doi
  56. (1977). Nonlinear ordinary differential equations. Oxford applied mathematics and computing science series. doi
  57. (1983). Nonlinear oscillations, dynamical systems and bifurcations of vector fields. doi
  58. Observation of chaotic dynamics of coupled nonlinear oscillators. doi
  59. On relaxation oscillations. doi
  60. On the cellular instability of flames near porous-plug burners. doi
  61. On the equation of a curved flame front. doi
  62. On the hydrodynamic stability of curved premixed flames. doi
  63. (1979). On the symmetry- breaking bifurcation of chaotic attrac-
  64. (1944). On the theory of slow combustion. doi
  65. (1989). Parallel running system of three oscillators coupled through a six-port magic junction. doi
  66. (1987). Partial differential equations. doi
  67. (1987). Perturbation methods, bifurcation theory and computer algebra, doi
  68. Phase dynamics of weakly unstable periodic structures. doi
  69. Phase locking of Josephson junction series arrays. doi
  70. Phase portraits and bifurcations of a non-linear oscillator. doi
  71. Quasiperiodicity and chaos in a system with three competing frequencies. doi
  72. (1991). Rotation sets and mode locking in a three oscillator system. doi
  73. (1989). Scaling laws and bifurcation. In Singularity theory and its applications, doi
  74. (1989). Secondary infinite-period bifurcation of spinning combustion waves near a hydrodynamic cellular stability boundary. Sandia Report 89-8660, Sandia National Labs, doi
  75. (1990). Slow dynamics in a globally coupled oscillator array. doi
  76. Smooth functions invariant under the action of a compact Lie group. doi
  77. Some global dynamical properties of the Kuramoto-SiN-ashinsky equations: nonlinear stability and attractors. doi
  78. Spinning waves in gaseous combustion. doi
  79. Steady solutions of the Xuramoto-Sivashinsky equation. doi
  80. (1989). Steady-state mode interactions in rectangular domains. MSc thesis,
  81. Suppression of period doubling in symmetric systems. doi
  82. (1991). Symmetric chaos. doi
  83. Symmetry and phase locking in chains of weakly coupled oscillators. doi
  84. Symmetry breakdown from bifurcation. doi
  85. (1989). Symmetry breaking and the maximal isotropy subgroup conjecture for reflection groups. doi
  86. (1991). Symmetry- breaking in equivariant circle maps. doi
  87. Symmetry-increasing bifurcation of chaotic attractors. doi
  88. Symmetry-recovering crises of chaos in polarization-related optical bistability. doi
  89. (1991). Synchronisation of pulse-coupled biological oscillators.
  90. (1980). The Art of Electronics. doi
  91. The behavior of rings of coupled oscillators. doi
  92. The chemical basis of morphogenesis. doi
  93. (1992). The dynamics of n identical oscillators with symmetric coupling. doi
  94. (1980). The geometry of biological time, doi
  95. (1928). The heartbeat considered as a relaxation oscillator and an electrical model of the heart. doi
  96. (1976). The Hopf bifurcation and its applications, doi
  97. The Kuramoto-Sivashisky equation: a bridge between PDEs and dynamical systems. doi
  98. The modulated phase shift for stronly nonlinear, slowly varying and weakly damped oscillators. doi
  99. The steady states of the Kuramoto-Sivashinsky equation. doi
  100. (1945). The theory of sound, doi
  101. (1981). Theory and applications of Hopf bifurcation, volume 41 of LMS lecture notes. doi
  102. (1944). Theory of combustion and detonation of gases. doi
  103. (1982). Theory of laminar flames. Cambridge monographs on mechanics and applied mathematics. doi
  104. (1966). Theory of oscillators. Pergamon, doi
  105. Three identical oscillators with symmetric coupling. doi
  106. Transition to topological chaos for circle maps. doi

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