ARTTE Applied Researches in Technics, Technologies and Education Journal of the Faculty of Technics and Technologies, Trakia University https://sites.google.com/a/trakia-uni.bg/artte/ ARTTE Vol. 6, No. 3, 2018 ISSN 1314-8788 (print), ISSN 1314-8796 (online), doi: 10.15547/artte.2018.03.007 252 PHASE PORTRAIT OF SECOND-ORDER DYNAMIC SYSTEMS Kaloyan Yankov Faculty of Technics and Technologies of Yambol, Trakia University of Stara Zagora Graf Ignatiev 38, 8600 Yambol, Bulgaria e-mail: kaloyan.yankov@trakia-uni.bg Abstract. The phase portrait of the second and higher order differential equations presents in graphical form the behavior of the solution set without solving the equation. In this way, the stability of a dynamic system and its long-time behavior can be studied. The article explores the capabilities of Mathcad for analysis of systems by the phase plane method. A sequence of actions using Mathcad's operators to build phase portrait and phase trace analysis is proposed. The approach is illustrated by a model of plasma renin activity after treatment of experimental animals with nicardipine. The identified process is a differential equation of the second order. The algorithm is also applicable to systems of higher order. Keywords: phase portrait, phase trajectory, equilibrium point, stability analysis, bifurcation, Mathcad. “That which is static and repetitive is boring. That which is dynamic and random is confusing. In between lies art.” John A. Locke (1632–1704), British philosopher and medical researcher 1. INTRODUCTION The dynamic behavior of a system is represented by the ordinary differential equation (ODE) of order n: ) ( ) ( ... ) ( ) ( ) ( 0 2 2 2 1 1 1 t u b t y a dt t y d a dt t y d a dt t y d n n n n n n n          (1) One approach to studying the model (1) is the analytical solution. For linear systems, there are algorithms, making the task solvable. But this is not the same with non-linear systems. Their decision may be a problem because there are no common methods for solving nonlinear differential equations. As a rule, these systems are examined qualitatively, approximately [1,2]. Another approach, which is characterized by visual outcomes, is the method of phase space. It basically relies on graphical images representing different aspects of the differential model. In this way, special points can be found in which the system is stable or unstable, as well as tracking the model's long-time behavior and gaining information on the stability of the system. The present work aims to propose an algorithm for analysis of a system of order n using the phase space method in the environment of Mathcad, Parametric Technology Corporation [3, 4]. The capabilities of the program for symbol processing and analytical transformation will be used.