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Dynamical systems sits at the interface of pure and applied mathematics, containing some beautiful theory, as well as applications in diverse fields including numerical analysis, biological systems, economics and medicine.It is often difficult or impossible to write down an exact solution to systems of nonlinear equations. The emphasis in this course will be on qualitative techniques for classifying the behaviour of nonlinear systems, without necessarily solving them exactly. Two main types of dynamical systemswill be studied: discrete systems, consisting of an iterated map; and continuous systems, consisting of nonlinear differential equations.Topics covered will include: bifurcations and chaotic behaviour of interval maps; symbolic dynamics; topological model of chaos; mass transport and probabilistic dynamics; phase portrait analysis (Hartman-Grobman theorem, hyperbolicity of limit cycles, invariant manifolds, global bifurcations); centre manifolds.This course is independent of MATH363 Dynamical Systems, although previous enrolmentthere is desirable.
To demonstrate a breadth of knowledge of dynamical systems theory traversing smooth, continuous and probabilistic settings and to apply that knowledge correctly in new situationsTo be able to articulate what it means for a dynamical system to be chaotic, including the relation to randomnessTo choose and apply appropriate theoretical and numerical tools to analyse a given dynamical system and communicate clear and correct explanations of its global asymptotic behaviourTo exhibit mastery of both the power and limitations of standard methods of linearisation, analysis via invariant manifolds and symbolic dynamicsTo evaluate critically the findings and discussions in relevant original literature, and to exhibit familiarity with content that is relevant to the syllabus, but sits outside itTo engage in rigorous investigation and analysis of problems in dynamical systems both independently and collaboratively
This course will provide students with an opportunity to develop the Graduate Attributes specified below:
Critically competent in a core academic discipline of their award
Students know and can critically evaluate and, where applicable, apply this knowledge to topics/issues within their majoring subject.
Employable, innovative and enterprising
Students will develop key skills and attributes sought by employers that can be used in a range of applications.
Subject to approval of the Head of School.
Students must attend one activity from each section.
School of Mathematics and Statistics Postgraduate Handbook
General information for students
Domestic fee $1,017.00
International Postgraduate fees
* All fees are inclusive of NZ GST or any equivalent overseas tax, and do not include any programme level discount or additional course-related expenses.
For further information see
Mathematics and Statistics