APPLIED NON-LINEAR DYNAMICS SYSTEMS

Course Information
TitleΕΦΑΡΜΟΣΜΕΝΑ ΜΗ-ΓΡΑΜΜΙΚΑ ΔΥΝΑΜΙΚΑ ΣΥΣΤΗΜΑΤΑ / APPLIED NON-LINEAR DYNAMICS SYSTEMS
Code227
FacultyEngineering
SchoolMechanical Engineering
Cycle / Level1st / Undergraduate
Teaching PeriodSpring
CoordinatorVasileios Rothos
CommonYes
StatusActive
Course ID600018290

Programme of Study: UPS of School of Mechanical Engineering

Registered students: 35
OrientationAttendance TypeSemesterYearECTS
EnergyElective Courses belonging to the other845
Design and StructuresElective Course belonging to the selected specialization (Elective Specialization Course)845
Industrial ManagementElective Courses belonging to the other845

Class Information
Academic Year2020 – 2021
Class PeriodSpring
Faculty Instructors
Instructors from Other Categories
Weekly Hours4
Class ID
600171094
Course Category
Knowledge Deepening / Consolidation
Mode of Delivery
  • Face to face
Digital Course Content
Erasmus
The course is also offered to exchange programme students.
Language of Instruction
  • Greek (Instruction, Examination)
  • English (Instruction, Examination)
Prerequisites
Required Courses
  • 102 PHYSICS
  • 105 INFORMATICS
  • 116 DYNAMICS
  • 124 MECHANICAL VIBRATION AND MACHINE DYNAMICS
  • 214 STRUCTURAL DYNAMICS
  • 101 CALCULUS I (MATHEMATICS I)
  • 106 CALCULUS II (MATHEMATICS II)
  • 111 DIFFERENTIAL EQUATIONS (MATHEMATICS III)
  • 120 NUMERICAL ANALYSIS
  • 131 LINEAR ALGEBRA
Learning Outcomes
In this course the student acquires the essential knowledge and technical skills for analyzing Dynamical Systems, i.e, systems of coupled ordinary differential equations, with emphasis on nonlinear equations. The student becomes acquainted with the analytical tools to correctly interpret the dynamical behavior in phase space, which is one of the central concepts in the field of Dynamical Systems. The course is structured in such a way that the level of complexity is increased gradually: the student is first introduced to Dynamical Systems which have a two-dimensional phase space, and later to systems with three or more dimensions, where the dynamics can become chaotic. The notion of chaos (i.e. sensitive dependence on initial conditions, also called "the butterfly effect") and its ubiquity in the natural world, in economics, in the health sciences and innumerable other applications of practical interest, is one of the major new insights of Mathematics of the past 50 years. After having successfully followed the course, the student will have acquired a sound working knowledge of Dynamical Systems and will be able to follow the new scientific developments in this active and highly relevant research field.
General Competences
  • Apply knowledge in practice
  • Retrieve, analyse and synthesise data and information, with the use of necessary technologies
  • Work in an international context
  • Work in an interdisciplinary team
  • Be critical and self-critical
Course Content (Syllabus)
Autonomous systems of ODEs in the two-dimensional phase plane, equilibrium points and their stability properties, the importance of the nonlinear terms. Population dynamics of two competing species (Lotka-Volterra model) and other applications. Hamiltonian dynamical systems, gradient systems. Local vs. global stability, Lyapunov functions. Periodic solutions, limit cycles and the Poincaré-Bendixson theorem. The Van der Pol oscillator and other applications. The notion of structural stability/instability. Bifurcations of equilibrium points and periodic trajectories: saddle-node, transcritical, pitchfork and Hopf bifurcations. Systems of ODEs with a phase space of three or more dimensions, the appearance of chaotic behavior. The Lorenz attractor and other chaotic ("strange") attractors in phase space. Study of stationary and travelling wave solution of non-linear PDEs, soliton equations (Korteweg-de Vries, Nonlinear Schrodinger & sine-Gordon). We study the existence and stability of solitons with methods of applied dynamical systems.
Educational Material Types
  • Notes
  • Slide presentations
  • Book
Use of Information and Communication Technologies
Use of ICT
  • Use of ICT in Course Teaching
  • Use of ICT in Communication with Students
Description
Projector and interactive learning process.
Course Organization
ActivitiesWorkloadECTSIndividualTeamworkErasmus
Lectures521.7
Reading Assigment401.3
Tutorial280.9
Written assigments270.9
Exams30.1
Total1505
Student Assessment
Description
Courseworks-Oral Examination
Student Assessment methods
  • Written Assignment (Formative, Summative)
  • Oral Exams (Formative, Summative)
Bibliography
Course Bibliography (Eudoxus)
Βιβλίο [3674]: ΔΥΝΑΜΙΚΑ ΣΥΣΤΗΜΑΤΑ ΚΑΙ ΧΑΟΣ ΤΟΜΟΣ Β΄, ΑΝΑΣΤΑΣΙΟΣ Χ. ΜΠΟΥΝΤΗΣ Βιβλίο [236]: ΕΦΑΡΜΟΣΜΕΝΑ ΜΑΘΗΜΑΤΙΚΑ, LOGAN DAVID J.
Additional bibliography for study
1. Jordan D W and P. Smith, Nonlinear Ordinary Differential Equations, 2nd Edition, Clarendon Press, Oxford, 1987. 2. E. Infeld & G. Rowlands, Nonlinear Waves Solitons and Chaos, Cambridge University Press, 2000. 3. T. Kapitula & K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, Springer 2013. 4. M.J. Ablowitz, Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Cambridge University Press, 2011. 5. R.V.Churchill and J.W.Brown, Μιγαδικές Συναρτήσεις και Εφαρμογές. Πανεπιστημιακές Εκδόσεις Κρήτης.
Last Update
10-05-2021