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Effectiveness of mathematical education as non-linear, composite and open system, formation and development of cognitive abilities of the trainee are wholly defined in the solution of complex tasks by means of modern achievements in science to high school practice adaptation. The possibility of complex tasks solution arises at identification of "problem zones" of mathematical education and creation of the generalized constructs of a substance of basic educational elements. As an example of a research of "problem zone" in high school mathematics the problem of visualization and a research of a phase portrait of decisions in the qualitative theory of differential equations for one or several variables functions in the continuous time is considered All stages of developed technology of synergetic effects manifestation are explicated on the example of composite educational construct development ― solution stability of differential equations system. As the didactic mechanism of generalized construct substance development the authors suggest a base of professional and applied tasks and modern information and communication technologies of different level: distant environments, mathematical packages, cross-platform environments. Through mathematical and computer modeling it is possible to build the stages of adaptation and technological constructs of updating and manifestation of synergy in mathematics training on the basis of cultures dialogue. The offered technology allows to enhance educational and professional motivations of students, development quality of mathematical knowledge, contributes to efficient development of intellectual thinking operations with the possibility of selfdevelopment and manifestation of creative independence of the person.

Technology of Synergy Manifestation in the Research of Solution’s Stability of Differential Equations System