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COMPLEX SYSTEM WITH FLOWS AND SYNCHRONIZATION

Bull. Cal. Math. Soc.，2017，109（6）：461–484 | 2017年12月05日

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A complex system S consists of m components, maybe inconsistence with m ≥ 2 such as those of self-adaptive systems, particularly the biological systems and usually, a system with contradictions, i.e., there are no a classical mathematical subfield applicable. Then, how can we hold its global and local behaviors or true face? All of us know that there always exists universal connections between things in the world, i.e., a topological graph −→G underlying parts in S. We can thereby establish mathematics over a graph family −→G1,−→G2, · · · for characterizing the dynamic behaviors of system S on the time t, i.e., complex flows. Formally, a complex flow −→G L is a topological graph −→G associated with a mapping L : (v, u) → L(v, u), 2 end-operators A+ vu : L(v, u) → LA + vu(v, u) and A+ uv : L(u, v) → LA + uv (u, v) on a Banach space B over a field F with L(v, u) = −L(u, v) and A+ vu(−L(v, u)) = −LA + vu(v, u) for ∀(v, u) ∈ E −→G holding with continuity equations dxv dt = u∈NG(v) LA + vu (v, u) , ∀v ∈ V −→G, where xv is a variable on vertex v for ∀v ∈ E −→G. Particularly, if dxv/dt = 0 for ∀v ∈ V −→G, such a complex flow −→G L is nothing else but an action flow or conservation flow. The main purpose of this lecture is to clarify the complex system with that of contradictory system and its importance to the reality of a thing T by extending Banach or Hilbert spaces to Banach or Hilbert continuity flow spaces over topological graphs −→G1,−→G2, · · · and establishing the global differential theory on complex flows, characterize the global dynamic behaviors of complex systems, particularly, complex networks independent on graphs, for instance the synchronization of complex systems by applying global differential on the complex flows −→G L .

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