MAT7104 Dynamical Systems

Dynamical systems describe the time evolution of systems which arise from physics, biology, chemistry and other areas. As mathematical objects they are ordinary differential equations, usually nonlinear and therefore not usually able to be explicitly solved. The aim of the course is to see how to make a qualitative analysis of a dynamical system using many different analytic tools. Thus, the course introduces the processes of modelling real engineering systems with discrete and continuous mechanical, fluid, thermal or electrical elements. It illustrates the development of the governing differential equations associated with dynamic mechanical systems. The course further defines the possible methods of deriving solutions to the systems governing equations as well as chaotic behavior of discrete dynamical systems. Orbit of a map configuration space and phase space are introduced. Bifurcations and stability concepts, nonlinear oscillators in Hamiltonian systems, randomness, chaos, strange attraction and mechanism of turbulence are discussed.

Course Objectives
Upon successful completion of this subject students should be able to:

• This course contributes primarily to the students' knowledge of engineering topics, and does provide design experience.
• The course intends to impart to graduate students the knowledge and skills related to basic ideas of nonlinear dynamical systems
• Solve simple problems in calculus and linear algebra, and apply this knowledge to solve selected problems arising in other fields of science, engineering, medicine, economics, finance and social sciences.
• Demonstrate knowledge of calculus and linear algebra and its theoretical underpinnings by constructing logical, clearly presented and justified arguments incorporating deductive reasoning.
• Apply multiple approaches to solve a simple problem in calculus or linear algebra to obtain comparable answers. Identify where basic mathematical concepts are useful in addressing a real-world problem.
• Engage in group discussions with peers to decide upon the appropriate choice of mathematics to describe a wide variety of problems from fields including the physical sciences, life sciences, engineering and business.
• Identify where basic mathematical concepts are useful in addressing a real-world problem.

Learning Outcomes
This subject also contributes specifically to the development of following course intended learning outcomes:

• Upon completion of this course the students will be able to analyse simple dynamical systems.
• Develop theoretical and technical knowledge in an area of the mathematical sciences, incorporating deductive reasoning to solve complex problems.
• Examine the principles and concepts of a range of fundamental areas in the mathematical sciences (calculus, discrete mathematics, linear algebra, probability, statistics and quantitative management).
• Make arguments based on proof and conduct simulations based on selection of approaches (e.g. analytic vs numerical/experimental, different statistical tests, different heuristic algorithms) and various sources of data and knowledge.
• Demonstrate professional and responsible analysis of real-life problems that require application of mathematics and statistics.
• Test critical thinking skills to create solutions for contemporary mathematical sciences problems.