This is an introduction to the qualitative theory of differential equations.

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The subject gives a mathematical background to the nonlinear control. This subject is also an introduction to partial differential equations of the second order. This web site uses cookies to deliver its users personalized dynamic content.

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Differential Equations and Stability Theory. Login Hrvatski hr English. Equilibrium points. Classification of phase portraits.

## Stability theory for a pair of trailing vortices

Nonlinear systems. Periodic solutions and limit cycles. Energy method. Ljapunov stability and instability theorems. Robustness of systems. Types and classification of partial differential equations of the second order.

## Stability theory for a pair of trailing vortices | AIAA Journal

Heat conduction equation. Wave equation. Laplace equation. General Competencies This is an introduction to the qualitative theory of differential equations. Learning Outcomes Define fundamental notions related to qualitative theory of ordinary differential equations and fundamental notions related to partial differential equations of second order. Relate knowledge achived in linear algebra to behaviour of linear and nonlinear systems Compare knowledge from other courses with mathematical knowledge about differential equations.

Recognize difference between analytic, numeric and qualitative approach to differential equations. Forms of Teaching Lectures Lectures. Exams Midexam, exam, seminar Consultations Consulting. Seminars seminar. Week by Week Schedule Eigenvalues and the Jordan form of a matrix. Linear systems of differential equations in the plane. Klasifikacija faznih portreta linearnih sustava oko Classification of phase portraits of linear systems near the equilibrium points. Stability and asymptotical stability.

Closed trajectory and limit cycle.

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Linearization, Hartman-Grobman theorem. Stability theory. Invariant manifolds. Bendixson theorem, Duffing equation. Poincare index theory Dissipative and Conservative systems, energy method.

## Theory of isolated magnetic skyrmions: From fundamentals to room temperature applications

Fundamentals of a Fuzzy-Logic-Based Generalized Theory of Stability Abstract: Stability is one of the fundamental concepts of complex dynamical systems including physical, economical, socioeconomical, and technical systems. In classical terms, the notion of stability inherently associates with any dynamical system and determines whether a system under consideration reaches equilibrium after being exposed to disturbances. Predominantly, this concept comes with a binary Boolean quantification viz.

While in some cases, this definition is well justifiable, with the growing complexity and diversity of systems one could seriously question the Boolean nature of the definition and its underlying semantics. This becomes predominantly visible in human-oriented quantification of stability in which we commonly encounter statements quantifying stability through some linguistic terms such as, e. To formulate human-oriented definitions of stability, we may resort ourselves to the use of a so-called precisiated natural language, which comes as a subset of natural language and one of whose functions is redefining existing concepts, such as stability, optimality, and alike.

Being prompted by the discrepancy of the definition of stability and the Boolean character of the concept itself, in this paper, we introduce and develop a generalized theory of stability GTS for analysis of complex dynamical systems described by fuzzy differential equations. Different human-centric definitions of stability of dynamical systems are introduced.

We also discuss and contrast several fundamental concepts of fuzzy stability, namely, fuzzy stability of systems, binary stability of fuzzy system, and binary stability of systems by showing that all of them arise as special cases of the proposed GTS. The introduced definitions offer an important ability to quantify the concept of stability using some continuous quantification that is through the use of degrees of stability.

In this manner, we radically depart from the previous binary character of the definition. We establish some criteria concerning generalized stability for a wide class of continuous dynamical systems. Next, we present a series of illustrative examples which demonstrate the essence of the concept, and at the same time, stress that the existing Boolean techniques are not capable of capturing the essence of linguistic stability.