We have investigated periodic and turbulent waves in excitable media and, in
particular, in the Belousov-Zhabotinsky
(BZ) reaction. We found that turbulence can be
induced by high light intesity or low catalyst concentrations (in the Ru-catalyzed
reaction) by oxygen, or by methanol.

The different spatiotemporal modes were analyzed by correlation analysis of video images, and they were simulated both by cellular automata (CA) and by partial differential equations. The figure below shows CA-simulations of three-dimensional waves.

There exist conditions for which a short light pulse can cause splitting into a forwards and a backwards running wave. If this is done with a spiral wave, the two resulting spirals annihilate each other.

Due to the formal analogy (analogous form of differential equations)
between the BZ reaction and heart muscle, BZ-turbulence is comparable to
the fatal heart fibrillation. Moreover, considering that light in the BZ-reaction
corresponds to electrical current in the heart, the annihilation
of spirals points to a method of controlling heart tachycardia. Formerly,
we investigated the physiological clock (of yeast) due to oscillating enzymatic
breakdown of sugar. Considering the coupling to membrane transport, one
obtains, under certain conditions, chaotic biorhythms.

Examples of 3D waves in an excitable medium (simulations)

**Publications**

B. Maiworm, M. Schmick and M. Markus, "Transduction of electrochemical into mechanical oscillations", Physics Letters A 361, 488-492 (2007)

B. Maiworm and M. Markus, "Mechanical oscillations of an oxidizing agent over a mercury surface". ScienceAsia 31, 329-333 (2005).

M. Woltering and M. Markus, "Annihilation of turbulence in excitable systems using activator-flux pulses", Physica D 168/9, 23-34 (2002)

M. Woltering and M. Markus, "Turbulence control by wave splitting in excitable media", Phys. Lett. A 297, 363-369 (2002)

M. Woltering and M. Markus, "Control of Spatiotemporal Disorder in an Excitable Medium", Science Asia 28, 43-48 (2002)

M. Woltering and M. Markus, "Oscillations and turbulence induced by an activating agent in an active medium", Phys. Rev. E 64, 045601/1-4 (2001)

M. Markus, G. Kloss and I. Kusch, "Disordered waves in an homogeneous, motionless excitable medium", Nature 371, 402-404 (1994)

M. Markus, "The BZ reaction as 'analog computer' for ubiquitous excitable media", Chem. Intelligencer 5(3), 48-50 (1999)

A. P.-Munuzuri, V. Perez-Villar and M. Markus, "Splitting of autowaves in an active medium", Phys. Rev. Lett. 79, 1941-1944 (1997)

M. Markus, Zs. Nagy-Ungvarai and B. Hess, "Phototaxis of spiral waves", Science 257, (1992), pp. 225-227

M. Markus and B. Hess, "Isotropic cellular automaton for modelling excitable media", Nature 347, 56-58 and cover (1990)

M. Woltering and M. Markus, "Riddled-like basins of transient chaos", Phys. Rev. Lett. 84(4), 630-633 (2000)

M. Woltering, R. Girnus and M. Markus, "Quantification of turbulence in the Belousov-Zhabotinsky reaction by monitoring wave tips", J. Phys. Chem. A 103, 4034-4037 (1999)

M. Markus and A. Czajka, "Oxygen-induced wavefront instabilities and disorder in the Belousov-Zhabotinsky reaction", Chem. Phys. Lett. 289, 35-40 (1998)

M. Woltering and M. Markus, "Riddled basins of the Belousov-Zhabotinsky reaction", Chem. Phys. Lett. 321 (5-6), 473-478 (2000)

M. Markus, T. Schulte and A. Czajka, "Disordered waves in an excitable medium consisting of two layers: catalytic gel and non-catalytic solution", Physical Review E 56, 1-4 (1997)

A.P. Munuzuri and M. Markus, "Simplest optical feedback loop: excitation waves and their mirror image", Physical Review E 55, 33-35 (1997)

A.P. Munuzuri and M. Markus, "Cellular Automata Model and Measurements of Autowave splitting", Int. J. Bifurcation & Chaos 6, 1837-1844 (1996)

M. Markus and K. Stavridis, "Observation of chemical turbulence in the Belousov-Zhabotinsky reaction", Journal of Bifurcation and Chaos 4, 1233-1243 (1994)

M. Markus and K. Stavridis, "Wavefront deformations in excitable media: chemical lenses and ripples", Phil. Trans. Roy. Soc. A 347 (1994), pp. 601-605

A. Holden, M. Markus and H. Othmer (eds.), "Nonlinear Wave Processes in Excitable Media", Plenum Press, New York (1991)

M. Markus, "Modelling morphogenetic processes in excitable tissues using novel cellular automata". Biomed. Biochim. Acta 49, 681-696 (1990)

M. Markus, S.C. Müller and G. Nicolis (eds.), "From chemical to biological organization", Springer-Verlag, Heidelberg (1988)

M. Markus, S. Müller, T. Plesser and B. Hess, "On the recognition of order and disorder", Biological Cybernetics, 57, 187-195 (1987)

M. Markus and B. Hess, "Order and chaos in biochemistry", Trends in Biochemical Sciences 12, 45-48 (1987)

M. Markus, D. Kuschmitz and B. Hess, "Properties of strange attractors in yeast glycolysis", Biophysical Chemistry 22, 95-105 (1985)

M. Markus and B. Hess, "Long-lived chaotic transients in glycolysis", in: 'Chronobiology and Chronomedicine' (ed. by G. Hildebrandt, R. Moog and F. Raschke), Verlag Peter Lang, Frankfurt, pp. 85-89 (1987)

M. Markus, E. Liefke and U. Onken, "A new chemical oscillator based on the oxidation of hydrazine", in: 'Complex chemical reaction systems: Mathematical modelling and simulations' (ed. by J. Warnatz und W. Jaeger) Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, pp. 160-168 (1987)

M. Markus, S.C. Müller, Th. Plesser and B. Hess,"Characterization of order and disorder in two-dimensional patterns", in: 'Chaos in Biological Systems' (ed. by H. Degn, A. Holden and L.F. Olsen) Plenum Press, N.Y. (1987) pp. 295-303

M. Markus and B. Hess, "Dimension and Lyapunov exponents of a strange attractor from biochemical data", in: 'Temporal Order' (L. Rensing and N.I. Jaeger, eds.) Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, pp. 191-193 (1985)

B. Hess and M. Markus, "Chemische Uhren" in 'Selbstorganisation-Die Entstehung von Ordnung in Natur und Gesellschaft' (A. Dress, H. Hendrichs and G. Kueppers, eds.) Piper-Verlag, Muenchen, pp. 61-79 (1986)

M. Markus and B. Hess, "Control of metabolic oscillations: unpredictability, critical slowing down, optimal stability and hysteresis" in 'Control of metabolic processes' (ed. by A. Cornish-Bowden) Plenum Press, New York (1990), pp. 303-313

M. Markus, "Self-organisation in chemistry and biology" in 'Instationary processes and dynamic experimental methods in catalysis, electrochemistry and corrosion' (ed. by G. Sandstede and G. Kreysa) VCH-Verlag, Weinheim (1989) pp. 445-466

M. Markus, "Quantitative characterization of complex oscillators", Habilitationsschrift, Univ. Dortmund (1988)

M. Markus and B. Hess, "Input-response relationships in the dynamcis of glycolysis", Arch. Biol. Med. Exp. 18, 261-271 (1985)

M. Markus, S.C. Müller and B. Hess, "Observation of entrainment, quasiperiodicity and chaos in glycolyzing yeast extract under periodic glucose input", Ber. Bunsenges. Phys. Chem. 89, 651-654 (1985)

B. Hess and M. Markus, "The diversity of biochemical time patterns", Ber. Bunsenges. Phys. Chem. 89, 642-651 (1985)

M. Markus, D. Kuschmitz and B. Hess, "Chaotic dynamics in yeast glycolysis under periodic substrate input flux", FEBS Lett. 172, 235-238 (1984)

M. Markus and B. Hess, Transitions between oscillatory modes in a glycolytic model system", Proc. Natl. Acad. Sciences USA 81, 4394-4398 (1984)