Recent work by Konstantin Mischaikow

Conley Index

• Survey Articles

• K. Mischaikow and M. Mrozek ,  Conley Index Theory. (The final version can be found in Handbook of Dynamical Systems II: Towards Applications, (B. Fiedler, ed.) North-Holland, 2002.
• K. Mischaikow, The Conley Index Theory: A Brief Introduction .  (The final version can be found in Conley Index Theory, Banach Center Publications, 47, 1999.)
• L. Arnold, C. Jones, K. Mischaikow, and G. Raugel, Dynamical Systems, Lecture Notes in Mathematics 1609 (R. Johnson ed.), Springer, 1995.
  
• The structure of isolated invariant sets and the Conley Index, Contemporary Mathematics (ed. C. McCord), 152(1993), 269-290.
  

• Structure of Invariant Sets

• Theory

• Maria C. Carbinatto, and K. Mischaikow, Horseshoes and the Conley Index Spectrum - II: The Theorem is Sharp (The final version can be found in Discrete and Continuous Dynamical Systems,5 (1999) 599-616)
• Maria C. Carbinatto, Jarek Kwapisz, and K. Mischaikow, Horseshoes and the Conley Index Spectrum (this is an update of the October 7, 1996 version) 
• C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Journal AMS 9(4)(1996), 1095-1133. 
• C. McCord, M. Mrozek, and K. Mischaikow, Zeta functions, periodic trajectories, and the Conley index, J. Diff. Eqns. 121(2)(1995), 258-292.  
• M. Mrozek, and K. Mischaikow,  Isolating neighborhoods and chaos, Japan J. Industrial & Applied Math., 12(2)(1995) 205-236.
  
• C. McCord, K. Mischaikow, and M. Mrozek, Zeta functions, periodic trajectories, and the Conley index, J. Diff. Eqns. 121(2)(1995), 258-292.
  

• Applications

• S. Day, O. Junge, and K. Mischaikow, A Rigorous Numerical Method for the Global Analysis of Infinite Dimensional Discrete Dynamical Systems  (To appear in SIAM Dynamical Systems).
• K. Mischaikow, M. Mrozek , and A. Szymczak ,  Chaos in the Lorenz equations: A computer assisted proof. Part III: Classical Case Parameter Values
• K. Mischaikow and M. Mrozek , Chaos in the Lorenz equations: A computer assisted proof. Part II: Details (The final version can be found in Mathematics of Computation, 67 (1998) 1023-1046) 
  
• T. Gedeon and K. Mischaikow, Structure of the global attractor of cyclic feedback systems, J. Dyn. & Diff. Eq., 7(1)(1995), 141-190.
  
• T. Gedeon and K. Mischaikow, Dynamics of cyclic feedback systems, Resenhas IME-USP, 1 (1994), 495-515
  
• Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal., 26(5)(1995), 1199-1224.
• K. Mischaikow and Y. Morita, Dynamics on the global attractor of a gradient flow arising from the Ginzburg-Landau equation, Japan J. of Indust. & Appl. Math., 11(1994), 185-202.
  
• K. Mischaikow and M. Mrozek, Chaos in the Lorenz equations: A computer-assisted proof, Bulletin AMS, 32(1)(1995), 66-72.
  

• Connection Matrices and Transition Matrices

• Theory

• H. Kokubu, K. Mischaikow, and H. Oka, Directional Transition Matrix.  (The final version can be found in Conley Index Theory, Banach Center Publications, 47, 1999.)
• K. Mischaikow and R. Franzosa, Algebraic transition matrices in the Conley Index theory, Trans. AMS, 350(3) (1998), 889-912.
  
• C. McCord and K. Mischaikow, On the global dynamics of attractors for scalar delay equations, Journal AMS 9(4) (1996), 1095-1133.
  
• C.  McCord and K. Mischaikow, Equivalence of topological and singular transition matrices in the Conley Index theory, Michigan Math. J. 42 (1995), 387-414.
  
• C.  McCord and K. Mischaikow, Connected simple systems, transition matrices, and heteroclinic bifurcations, Trans. AMS 333(1) (1992), 397-422.
• K. Mischaikow and R. Franzosa, The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces, J. Diff. Eqns., 71(2) (1988), 270-287.
• Existence of generalized homoclinic orbits for one parameter families of flows, Proc. Amer. Math. Soc., 103(1)(1988), 59-68.
  

• Applications

• Hiroshi Kokubu, K. Mischaikow, Yasumasa Nishiura, Hiroe Oka, and Takeshi Takaishi, Connecting orbit structure of monotone solutions in the shadow system (the final version can be found in JDE, 140 (1997) 309-364).
• H. Kokubu, K. Mischaikow, H. Oka, Existence of infinitely many connecting orbits in a singularly perturbed ordinary differential equation, Nonlinearity 9 (1996), 1263-1280.
  
• T. Gedeon and K. Mischaikow, Structure of the global attractor of cyclic feedback systems, J. Dyn. & Diff. Eq., 7(1)(1995), 141-190.
• Global asymptotic dynamics of gradient-like bistable equations, SIAM J. Math. Anal., 26(5) (1995), 1199-1224.
• K. Mischaikow and J. Reineck, Travelling waves in predator-prey systems, SIAM J. Math. Anal., 24 (1993), 1179-1214. 
• V. Hutson and K. Mischaikow,  Travelling waves for mutualist species, SIAM J. Math. Anal., 24 (1993), 987-1008.
  
• B. Fiedler and K. Mischaikow, Dynamics of bifurcations for variational problems with O(3) equivariance: A Conley index approach, Arch. Rat. Mech. Anal. 119 (1992), 145-196.
  
• H. Hattori and K. Mischaikow, A dynamical system approach to a phase transition problem, J. Diff. Eqns., 94 (2)(1991), 340-378.
• H. Hattori and K. Mischaikow, On the existence of intermediate magnetohydrodynamic shock waves, J. of Dyn. & Diff. Eqns., 2 (2)(1990), 163-175.
  

• Singular Perturbations
  

• Theory

• T. Gedeon and K. Mischaikow, Singular boundary value problems via the Conley index, (The final version can be found in Top. Meth. Nonlinear Anal., 28 (2006) 263-284) 
• T. Gedeon, H. Kokubu, K. Mischaikow, and H. Oka, The Conley index for fast-slow systems II: Higher-dimensional slow variable, (The final version can be found in Jour. Diff. Eqs., 225 (2006) 242-307) 
•  T. Gedeon, H. Kokubu, K. Mischaikow, H. Oka, and J. Reineck, The Conley index for fast-slow systems I: One dimensional slow variables. (The final version can be found in Jour. Dyn. Diff. Eqs., 11 (1999) 427-470)
• K. Mischaikow, M. Mrozek, and J. Reineck, Singular Index Pairs (The final version can be found in Jour. Dyn. Diff. Eqs., 11 (1999) 399-425)
  

• Applications

• M. Gameiro , T. Gedeon , W. Kalies, H. Kokubu, K. Mischaikow, and H. Oka,  Topological Horseshoes of Travelling Waves for a Fast-Slow Predator-Prey System (submitted).
• T. Gedeon , H. Kokubu, K. Mischaikow, and H. Oka,  Chaotic solutions in slowly varying perturbations of Hamiltonian systems with applications to shallow water sloshing (The final version can be found in J. Dynamics and Differential Equations, 14, 2002).
• V. Hutson and K. Mischaikow, Periodic Travelling Waves . (The final version can be found in Conley Index Theory, Banach Center Publications, 47, 1999.) 
• V. Hutson and K. Mischaikow, Singular limits for travelling waves for a pair of equations, Proc. Royal Society Edinburgh, 126 (1996), 399-411.
  

• Time Series

• Theory

• K. Mischaikow, M. Mrozek , A. Szymczak , and J. Reiss , From Time Series to Symbolic Dynamics: An Algebraic Topological Approach (preliminary version)

• Applications

• K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, Construction of Symbolic Dynamics from Experimental Time Series (The final version can be found in Physical Review Letters Vol. 82, No. 6 page 1144, February 1999).