The “hidden” dynamics of the Rössler attractor
Tài liệu tham khảo
Verhulst, 2005
O’Malley, 1974
Kevorkian, 1981
Wilcox, 1995
Hirsch, 1977
Nipp, 1985, Invariant manifolds of singularly perturbed ordinary differential equations, Z. Angew. Math. Phys., 36, 309, 10.1007/BF00945464
Hadjinicolaou, 1998, Asymptotic solution of stiff PDEs with the CSP method: the reaction diffusion equation, SIAM J. Sci. Comput., 20, 781, 10.1137/S1064827596303995
Goussis, 2006, Model reduction and physical understanding of slowly oscillating processes: the circadian cycle, SIAM Multiscale Model. Simul., 5, 1297, 10.1137/060649768
Kourdis, 2013, Algorithmic asymptotic analysis of the NF-κB signaling system, Comput. Math. Appl., 65, 1516, 10.1016/j.camwa.2012.11.004
Goeke, 2013, Quasi-steady state: searching for and utilizing small parameters, vol. 35, 153
Tikhonov, 1952, Systems of differential equations containing a small parameter multiplying the derivative, Mat. Sb., 31, 575
Levinson, 1950, Small periodic perturbations of an autonomous system with a stable orbit, Ann. of Math., 52, 727, 10.2307/1969445
Vasil’eva, 1963, Asymptotic behavior of solutions to certain problems involving nonlinear differential equations containing a small parameter multiplying the highest derivatives, Russian Math. Surveys, 18, 13, 10.1070/RM1963v018n03ABEH001137
Fenichel, 1979, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31, 53, 10.1016/0022-0396(79)90152-9
Jones, 1977
Kaper, 1999, An introduction to geometric methods and dynamical systems theory for singular perturbation problems, vol. 56, 85
Lam, 1988, Understanding complex chemical kinetics with computational singular perturbation, Proc. Combust. Inst., 22, 931, 10.1016/S0082-0784(89)80102-X
Maas, 1992, Simplifying chemical kinetics: intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88, 239, 10.1016/0010-2180(92)90034-M
Adrover, 2006, Natural tangent dynamics with recurrent biorthonormalizations: a geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets, Physica D, 213, 121, 10.1016/j.physd.2005.05.021
Roussel, 1990, Geometry of the steady-state approximation: perturbation and accelerated convergence methods, J. Chem. Phys., 93, 1072, 10.1063/1.459171
Gorban, 2003, Method of invariant manifold for chemical kinetics, Chem. Eng. Sci., 58, 4751, 10.1016/j.ces.2002.12.001
Gear, 2005, Constraint-defined manifolds: a legacy code approach to low-dimensional computation, J. Sci. Comput., 25, 17, 10.1007/s10915-004-4630-x
Contou-Carrere, 2008, Model reduction and control of multi-scale reaction-convection processes, Chem. Eng. Sci., 63, 4012, 10.1016/j.ces.2008.04.035
Ginoux, 2014, Flow curvature manifolds for shaping chaotic attractors: I. Rössler-like systems, J. Phys. A: Math. Theor., 42, 10.1088/1751-8113/42/28/285101
Kaper, 2002, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165, 66, 10.1016/S0167-2789(02)00386-X
Zagaris, 2004, Fast and slow dynamics for the computational singular perturbation method, Multiscale Model. Simul., 2, 613, 10.1137/040603577
Zagaris, 2005, Two perspectives on reduction of ordinary differential equations, Math. Nachr., 278, 1629, 10.1002/mana.200410328
Goussis, 2006, An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems, J. Comput. Phys., 214, 316, 10.1016/j.jcp.2005.09.019
Goussis, 2011, Model reduction for combustion chemistry, Fluid Mech. Appl., 95, 193
Lam, 1994, CSP method for simplifying kinetics, Int. J. Chem. Kinet., 26, 461, 10.1002/kin.550260408
Sanders, 2007
Goussis, 2013, The role of slow system dynamics in predicting the degeneracy of slow invariant manifolds: the case of vdP relaxation-oscillations, Physica D, 248, 16, 10.1016/j.physd.2012.12.013
Rössler, 1976, An equation for continuous chaos, Phys. Lett., 57A, 397, 10.1016/0375-9601(76)90101-8
Rössler, 1977, Chaos in abstract kinetics: two prototypes, Bull. Math. Biol., 39, 275, 10.1007/BF02462866
Mease, 2003, Timescale analysis for nonlinear dynamical systems, J. Guid. Control Dyn., 26, 318, 10.2514/2.5049
Collet, 2006
Guckenheimer, 2006, Chaotic attractors of relaxation oscillators, Nonlinearity, 19, 701, 10.1088/0951-7715/19/3/009
Ginoux, 2009
Chiba, 2011, Periodic orbits and chaos in fast–slow systems with Bogdanov–Takens type fold points, J. Differential Equations, 250, 112, 10.1016/j.jde.2010.09.022
Abramov, 2012, Suppression of chaos at slow variables by rapidly mixing fast dynamics through linear energy-preserving coupling, Commun. Math. Sci., 10, 595, 10.4310/CMS.2012.v10.n2.a9
Barrio, 2014, Unbounded dynamics in dissipative flows: Rössler model, Chaos, 24, 024407, 10.1063/1.4871712
Ginoux, 2014, The slow invariant manifold of the Lorenz–Krishnamurthy model, Qual. Theory Dyn. Syst., 13, 19, 10.1007/s12346-013-0104-6
Lam, 1991, Conventional asymptotics and computational singular perturbation for simplified kinetics modelling, vol. 384, 227
Jackson, 1990
Strogatz, 1994
Ruelle, 1971, On the nature of turbulence, Comm. Math. Phys., 20, 167, 10.1007/BF01646553
Verhulst, 2005, Invariant manifolds in dissipative dynamical systems, Acta Appl. Math., 87, 229, 10.1007/s10440-005-1159-4
Chicone, 2006
Valorani, 2005, Higher order corrections in the approximation of low-dimensional manifolds and the construction of simplified problems with the CSP method, J. Comput. Phys., 209, 754, 10.1016/j.jcp.2005.03.033
Goussis, 1992, A study of homogeneous methanol oxidation kinetics using CSP, Symp. (Int.) Combust., 24, 113, 10.1016/S0082-0784(06)80018-4
Valorani, 2003, CSP analysis of a transient flame-vortex interaction: time scales and manifolds, Combust. Flame, 134, 35, 10.1016/S0010-2180(03)00067-1
Goussis, 2012, Quasi steady state and partial equilibrium approximations: their relation and their validity, Combust. Theory Model., 16, 869, 10.1080/13647830.2012.680502
Wolfram Research Inc., Mathematica, Version 7.0, Champaign, IL, 2008.
Crutchfield, 1980, Power spectral analysis of a dynamical system, Phys. Lett. A, 76, 1, 10.1016/0375-9601(80)90130-9