On a simple case of possible non-deterministic chaotic behavior in compartment theory of biological observables

Chaos, Solitons and Fractals

Volumn 22, Issue 2, 2004, Pages 277-284

  Conte, Elio ^a   Federici, Antonio ^a   Zbilut, Joseph P ^b  

^a UNIVERSITY OF BARI   (Italy)

^b RUSH MEDICAL COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION THEORY; DIFFERENTIAL EQUATIONS; EQUATIONS OF MOTION; MATHEMATICAL MODELS; NONLINEAR SYSTEMS; OPTIMIZATION; PERTURBATION TECHNIQUES; PHYSIOLOGY;

COMPARTMENT THEORY; LIPSCHITZ CONDITIONS;

CHAOS THEORY;

EID: 1842832078     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.chaos.2003.10.036     Document Type: Article

References (21)

1. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
2. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
3. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
4. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
5. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
6. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
7. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
8. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
9. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
10. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
11. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
12. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
13. The relevance of singularities in physiological systems was suggested for the first time by J.P. Zbilut in collaboration with A. Hubler and C.L. Webber at the Conference: Fluctuations and Order: the New Synthesis (Conference sponsored by the Center for Nonlinear Studies, Loas Alamos National Laboratories and the Santa Fe Institute, Santa Fe, New Mexico, Sept. 9-12, 1993) by a contribution entitled "Nondeterministic Equations of Motion as a Result of Noise Perturbation." The paper was subsequently published by J.P. Zbilut, A. Hubler, C.L.Webber, Physiological Singularities Modeled by Nondeterministic Equations of Motion and the Effect of Noise, in M. Millonas editor, Fluctuations and Order: The New Synthesis, Springer Verlag, New York, 397-417, 1996. Other relevant published works include Zbilut J.P., Zak M., Webber C.L. Nondeterministic chaos approach to neural intelligence. intelligent engineering systems through artificial neural networks. vol. 4:1994;ASME Press, New York. p. 819-24 Zbilut J.P., Zak M., Webber C.L. Physiological singularities in respiratory and cardiac dynamics. Chaos, Solitons & Fractals. 5:1995;1509-1516 Zbilut J.P., Zak M., Meyers E.R. A terminal dynamics model of the heartbeat. Biologic. Cybernet. 75:1996;277-280 Zak M, Zbilut JP, Meyers ER. From instability to intelligence complexity and predictability in nonlinear dynamics, Lecture notes in physics: New Series m49, Berlin, Heidelberg, New York: Springer Verlag; 1997 Zbilut J.P., Zaldivar-Comenges J.-M., Strozzi F. Recurrence quantification based-Lyapunov exponents for monitoring divergence in experimental data. Phys. Lett. A. 297:2002;173-181 Zbilut J.P., Dixon D.D., Zak M. Detecting singularities of piecewise deterministic (terminal) dynamics in experimental data. Phys. Lett. A. 304:2002;95-101 Zak M. The problem of irreversibility in Newtonian dynamics. Int. J. Theoret. Phys. 31(2):1992;333-342 Zak M. Terminal chaos for information processing in neurodynamics. Biologic. Cybernet. 64:1991;343-351 Zak M. Non-Lipschitzian dynamics for neural net modelling. Appl. Math. Lett. 2(1):1989;69-74 Zbilut J.P., Thomasson N., Webber C.L. Recurrence quantification analysis as a tool for nonlinear exploration of nonstationarity cardiac signals. Med. Eng. Phys. 9:2001;1-8 Zak M. Dynamical simulation of probabilities. Chaos, Solitons & Fractals. 5:1997;793-804 Zak M. Non-Lipschitz Approach to Quantum Mechanics. Chaos, Solitons & Fractals. 9:1998;1183-1198.
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