*W. Otto Friesen and Jonathon A. Friesen*

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780195371833
- eISBN:
- 9780199865178
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195371833.003.0024
- Subject:
- Psychology, Cognitive Neuroscience

This chapter describes the method for numerical integration of the equations that underlie NeuroDynamix II models.

This chapter describes the method for numerical integration of the equations that underlie *NeuroDynamix II* models.

*David P. Feldman*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0029
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on differential equations, dynamical systems that change continuously, with the variable of interest having a value at every instant. It first provides an overview of discrete ...
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This chapter focuses on differential equations, dynamical systems that change continuously, with the variable of interest having a value at every instant. It first provides an overview of discrete dynamical systems and continuous change and then illustrates the use of the instantaneous rate of change to describe and understand quantities that change continuously. It also considers an example in which the Euler's method is used to approximate the solution to a differential equation.Less

This chapter focuses on differential equations, dynamical systems that change continuously, with the variable of interest having a value at every instant. It first provides an overview of discrete dynamical systems and continuous change and then illustrates the use of the instantaneous rate of change to describe and understand quantities that change continuously. It also considers an example in which the Euler's method is used to approximate the solution to a differential equation.

*David P. Feldman*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199566433
- eISBN:
- 9780191774966
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199566433.003.0031
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with two-dimensional differential equations and examines whether they are capable of a richer set of behaviours compared with their one-dimensional counterparts. It begins by ...
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This chapter deals with two-dimensional differential equations and examines whether they are capable of a richer set of behaviours compared with their one-dimensional counterparts. It begins by considering a model of two interacting populations, known as the Lotka-Volterra model or the Lotka-Volterra equations, a standard example of interacting populations in mathematical ecology. The example involves two populations of different creatures, perhaps rabbits and foxes, a two-dimensional system in the sense that both animal populations are both unknown functions. The chapter also illustrates how to adapt Euler's method to solve coupled equations, summarises the solutions to one-dimensional differential equations with a phase line, and discusses phase space and phase portraits. It concludes by describing the van der Pol equation and two types of stable, attracting behavior: a fixed point and a limit cycle.Less

This chapter deals with two-dimensional differential equations and examines whether they are capable of a richer set of behaviours compared with their one-dimensional counterparts. It begins by considering a model of two interacting populations, known as the Lotka-Volterra model or the Lotka-Volterra equations, a standard example of interacting populations in mathematical ecology. The example involves two populations of different creatures, perhaps rabbits and foxes, a two-dimensional system in the sense that both animal populations are both unknown functions. The chapter also illustrates how to adapt Euler's method to solve coupled equations, summarises the solutions to one-dimensional differential equations with a phase line, and discusses phase space and phase portraits. It concludes by describing the van der Pol equation and two types of stable, attracting behavior: a fixed point and a limit cycle.

*S. G. Rajeev*

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805021
- eISBN:
- 9780191843136
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805021.003.0015
- Subject:
- Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric ...
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Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric Integrators that respect these symmetries. Extending thesemethods to Euler and Navier-Stokes is an outstanding research problem in fluid mechanics. Therefore, a short review of geometric integrators for ODEs is given in this last chapter. Exponential coordinates on a Lie group are explained; the formula for differentiating a matrix exponential is given and used to derive the first few terms of the Magnus expansion. Geometric integrators corresponding to the Euler and trapezoidal methods for ODEs are given.Less

Generic methods for solving ordinary differential equations (ODEs, e.g., Runge-Kutta) can break the symmetries that a particular equation might have. Lie theory can be used to get Geometric Integrators that respect these symmetries. Extending thesemethods to Euler and Navier-Stokes is an outstanding research problem in fluid mechanics. Therefore, a short review of geometric integrators for ODEs is given in this last chapter. Exponential coordinates on a Lie group are explained; the formula for differentiating a matrix exponential is given and used to derive the first few terms of the Magnus expansion. Geometric integrators corresponding to the Euler and trapezoidal methods for ODEs are given.