Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach
Abstract
The modern dynamical systems of various physical natures, such as natural, social, economic, and technical ones, are complexes of various subsystems. They are connected by processes of intensive dynamic interaction and exchange of energy, matter, and information and incorporate nonlinear dynamics, memory, complicated transients, bifurcation and chaotic motion modes. Particularly, synergetics as a very young discipline deals with complex systems, i.e. it is concerned with the spontaneous formation of macroscopic spatial, temporal, or functional structures of systems via self-organization and is associated with systems composed of many subsystems, which may be of quite different natures. Synergetics take into account deterministic processes as treated in the dynamic systems theory including bifurcation theory, catastrophy theory, as well as basic notions of the chaos theory and develops its own approaches. Here, the fundamental basis of nonlinear theory of system's synthesis based on syn...ergetics as well as fractional calculus approach in modern control theory together with its application will be presented. The difference of synergetic approach from the classical scientific methods is in identification of the fundamental role of self-organization in nonlinear dynamic systems and it is necessary to keep the conceptual correspondence to the main qualities of self-organization: nonlinearity open systems coherence. Synergetic approach is based on the natural homeostatic-conservation of the internal qualities of the dynamic systems of various natures. Namely, Russian scientist AA. Kolesnikov developed a novel synergetic approach based on the ideas of modern mathematics, cybernetics, and synergetics to the synthesis of control systems for nonlinear, multidimensional and multilinked dynamic systems of various natures. The synergetic approach to control theory (synergetic control theory-SCT) is a novel nonlinear control method where the nonlinearities of a system are considered in the control design and a systematic design procedures. The invariants (synergies) and attractors, introduced as the main element of SCT, allow establishing direct link to the energy conservation laws, i.e. to the fundamental qualities of various objects. So, invariants, self-organization, and cascade synthesis are the fundamental notions of the SCT determining its essence, novelty, and content. Also, fractional calculus (FC) has a long history of three hundred years, over which a firm theoretical foundation has been established. All fractional operators consider the entire history of the process being considered, thus being able to model the non-local and distributed effects often encountered in natural and technical phenomena and they provide an excellent instrument for description of the memory, heredity, non-locality, self-similarity, and stochasticity of various materials and processes. Fractional dynamics can be encountered in various nonlinear dynamical systems such as viscoelastic materials, electrochemical processes, thermal systems, transmission and acoustics, chaos and fractals, biomechanical systems, and many others. The fractional dynamic systems with nonlinear control represent a relatively new class of applications of the FC which certified the FC as being a fundamental tool in describing the dynamics of complex systems as well as in advanced nonlinear control theory.
Keywords:
Synergetics / Self-organization / Phenomenology / Nonlinear systems / Invariants / Fractional calculusSource:
International Journal of Non-Linear Mechanics, 2015, 73, 31-42Publisher:
- Pergamon-Elsevier Science Ltd, Oxford
Funding / projects:
- Sustainability and improvement of mechanical systems in energetic, material handling and conveying by using forensic engineering, environmental and robust design (RS-MESTD-Technological Development (TD or TR)-35006)
- Development of methods and techniques for early diagnostic of cervical, colon, oral cavity cancer and melanoma based on a digital image and excitation-emission spectrum in visible and infrared domain (RS-MESTD-Integrated and Interdisciplinary Research (IIR or III)-41006)
DOI: 10.1016/j.ijnonlinmec.2014.11.011
ISSN: 0020-7462
WoS: 000355022100006
Scopus: 2-s2.0-84939962183
Collections
Institution/Community
Mašinski fakultetTY - JOUR AU - Lazarević, Mihailo PY - 2015 UR - https://machinery.mas.bg.ac.rs/handle/123456789/2240 AB - The modern dynamical systems of various physical natures, such as natural, social, economic, and technical ones, are complexes of various subsystems. They are connected by processes of intensive dynamic interaction and exchange of energy, matter, and information and incorporate nonlinear dynamics, memory, complicated transients, bifurcation and chaotic motion modes. Particularly, synergetics as a very young discipline deals with complex systems, i.e. it is concerned with the spontaneous formation of macroscopic spatial, temporal, or functional structures of systems via self-organization and is associated with systems composed of many subsystems, which may be of quite different natures. Synergetics take into account deterministic processes as treated in the dynamic systems theory including bifurcation theory, catastrophy theory, as well as basic notions of the chaos theory and develops its own approaches. Here, the fundamental basis of nonlinear theory of system's synthesis based on synergetics as well as fractional calculus approach in modern control theory together with its application will be presented. The difference of synergetic approach from the classical scientific methods is in identification of the fundamental role of self-organization in nonlinear dynamic systems and it is necessary to keep the conceptual correspondence to the main qualities of self-organization: nonlinearity open systems coherence. Synergetic approach is based on the natural homeostatic-conservation of the internal qualities of the dynamic systems of various natures. Namely, Russian scientist AA. Kolesnikov developed a novel synergetic approach based on the ideas of modern mathematics, cybernetics, and synergetics to the synthesis of control systems for nonlinear, multidimensional and multilinked dynamic systems of various natures. The synergetic approach to control theory (synergetic control theory-SCT) is a novel nonlinear control method where the nonlinearities of a system are considered in the control design and a systematic design procedures. The invariants (synergies) and attractors, introduced as the main element of SCT, allow establishing direct link to the energy conservation laws, i.e. to the fundamental qualities of various objects. So, invariants, self-organization, and cascade synthesis are the fundamental notions of the SCT determining its essence, novelty, and content. Also, fractional calculus (FC) has a long history of three hundred years, over which a firm theoretical foundation has been established. All fractional operators consider the entire history of the process being considered, thus being able to model the non-local and distributed effects often encountered in natural and technical phenomena and they provide an excellent instrument for description of the memory, heredity, non-locality, self-similarity, and stochasticity of various materials and processes. Fractional dynamics can be encountered in various nonlinear dynamical systems such as viscoelastic materials, electrochemical processes, thermal systems, transmission and acoustics, chaos and fractals, biomechanical systems, and many others. The fractional dynamic systems with nonlinear control represent a relatively new class of applications of the FC which certified the FC as being a fundamental tool in describing the dynamics of complex systems as well as in advanced nonlinear control theory. PB - Pergamon-Elsevier Science Ltd, Oxford T2 - International Journal of Non-Linear Mechanics T1 - Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach EP - 42 SP - 31 VL - 73 DO - 10.1016/j.ijnonlinmec.2014.11.011 ER -
@article{ author = "Lazarević, Mihailo", year = "2015", abstract = "The modern dynamical systems of various physical natures, such as natural, social, economic, and technical ones, are complexes of various subsystems. They are connected by processes of intensive dynamic interaction and exchange of energy, matter, and information and incorporate nonlinear dynamics, memory, complicated transients, bifurcation and chaotic motion modes. Particularly, synergetics as a very young discipline deals with complex systems, i.e. it is concerned with the spontaneous formation of macroscopic spatial, temporal, or functional structures of systems via self-organization and is associated with systems composed of many subsystems, which may be of quite different natures. Synergetics take into account deterministic processes as treated in the dynamic systems theory including bifurcation theory, catastrophy theory, as well as basic notions of the chaos theory and develops its own approaches. Here, the fundamental basis of nonlinear theory of system's synthesis based on synergetics as well as fractional calculus approach in modern control theory together with its application will be presented. The difference of synergetic approach from the classical scientific methods is in identification of the fundamental role of self-organization in nonlinear dynamic systems and it is necessary to keep the conceptual correspondence to the main qualities of self-organization: nonlinearity open systems coherence. Synergetic approach is based on the natural homeostatic-conservation of the internal qualities of the dynamic systems of various natures. Namely, Russian scientist AA. Kolesnikov developed a novel synergetic approach based on the ideas of modern mathematics, cybernetics, and synergetics to the synthesis of control systems for nonlinear, multidimensional and multilinked dynamic systems of various natures. The synergetic approach to control theory (synergetic control theory-SCT) is a novel nonlinear control method where the nonlinearities of a system are considered in the control design and a systematic design procedures. The invariants (synergies) and attractors, introduced as the main element of SCT, allow establishing direct link to the energy conservation laws, i.e. to the fundamental qualities of various objects. So, invariants, self-organization, and cascade synthesis are the fundamental notions of the SCT determining its essence, novelty, and content. Also, fractional calculus (FC) has a long history of three hundred years, over which a firm theoretical foundation has been established. All fractional operators consider the entire history of the process being considered, thus being able to model the non-local and distributed effects often encountered in natural and technical phenomena and they provide an excellent instrument for description of the memory, heredity, non-locality, self-similarity, and stochasticity of various materials and processes. Fractional dynamics can be encountered in various nonlinear dynamical systems such as viscoelastic materials, electrochemical processes, thermal systems, transmission and acoustics, chaos and fractals, biomechanical systems, and many others. The fractional dynamic systems with nonlinear control represent a relatively new class of applications of the FC which certified the FC as being a fundamental tool in describing the dynamics of complex systems as well as in advanced nonlinear control theory.", publisher = "Pergamon-Elsevier Science Ltd, Oxford", journal = "International Journal of Non-Linear Mechanics", title = "Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach", pages = "42-31", volume = "73", doi = "10.1016/j.ijnonlinmec.2014.11.011" }
Lazarević, M.. (2015). Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach. in International Journal of Non-Linear Mechanics Pergamon-Elsevier Science Ltd, Oxford., 73, 31-42. https://doi.org/10.1016/j.ijnonlinmec.2014.11.011
Lazarević M. Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach. in International Journal of Non-Linear Mechanics. 2015;73:31-42. doi:10.1016/j.ijnonlinmec.2014.11.011 .
Lazarević, Mihailo, "Elements of mathematical phenomenology of self-organization nonlinear dynamical systems: Synergetics and fractional calculus approach" in International Journal of Non-Linear Mechanics, 73 (2015):31-42, https://doi.org/10.1016/j.ijnonlinmec.2014.11.011 . .