Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

## MATH4A2 Techniques in Applied Mathematics 2

 Second Semester
10 points

This paper is an introduction to techniques for solving problems in applied mathematics. The focus of MATH4A2 will be on understanding the behavior of systems of equations. In many realistic applications, there are more than one dependent variables, and in this paper we will focus on techniques for studying systems of ordinary or partial differential equations. For systems of ordinary differential equations this will involve a study of dynamical systems and nonlinear dynamics, while we will also touch on chaos theory. Networks are becoming of increasing importance in applied mathematics, and we will outline some of the basic topics in network science, and then study dynamical systems which evolve over networks. We will also touch on the topic of stochastic differential equations. Returning to partial differential equations (PDEs), we will discuss how to solve systems of PDEs. We will also describe how certain PDE systems may result in pattern formation, or even turbulence.

### Prerequisites

MATH4A1, or permission of the lecturer.

### Topics

I. Systems of ordinary differential equations; discrete versus continuous time dynamical systems, steady states and linear stability analysis; dynamical systems arising in real-world applications; Transcritical bifurcation; Saddle-node bifurcation; Pitchfork bifurcation

II. Planar dynamics; phase plane analysis; periodic orbits; Hopf bifurcation; limit cycles

III. Nonlinear dynamics and chaos; phase portraits in more than two variables; volume contraction; properties of chaotic attractors in continuous-time dynamical systems; finding and simulating chaos in nonlinear systems

IV. A brief introduction to network science; dynamics on networks; coupled oscillators

V. Stochastic versus chaotic; the Itô stochastic integral; Stochastic differential equations (SDEs); exact solution of linear SDEs; Itô's Lemma for nonlinear SDEs

VI. Complications to linear PDEs seen in MATH4A1; non-autonomous partial differential equations; spatial heterogeneity; systems of PDEs

VII. Nonlinear PDEs; nonlinear waves, traveling waves, and solitons; instabilities and pattern formation; Turing instability; Benjamin-Feir instability; Turbulence; fascinating open problems and areas for future research

### Lecturers

Dr R. A. Van Gorder

You will be responsible for all material covered in lecture, but no other references are needed. I may provide additional optional references for specific topics, as appropriate.

### Assessment

Assessment will be through assignments only. These will be take-home problem sets.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

F = A

where:

• A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

Academic integrity means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

Academic misconduct is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

#### Plagiarism

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

• Although not intended, unintentional plagiarism is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided.
• Intentional plagiarism is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

#### Unauthorised Collaboration

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

#### Impersonation

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

#### Falsiﬁcation

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

#### Use of Unauthorised Materials

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

#### Assisting Others to Commit Academic Misconduct

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

Further information

While we strive to keep details as accurate and up-to-date as possible, information given here should be regarded as provisional. Individual lecturers will confirm teaching and assessment methods.