# Associate Professor Tim Langtry

**Associate Professor,**School of Mathematical Sciences

BAppSc (NSWIT), BA (Hons) (UNSW), MAppSc (NSWIT), PhD (UNSW)

Member, Australian Mathematical Society

**Phone**

+61 2 9514 2258

**Room**

CB01.15.37

### Research Interests

My main research interests are in numerical analysis and computational mathematics. Particular areas of interest include quasi-Monte Carlo methods for numerical multiple integration (eg rank 1 lattice rules) and, more recently, computational photonics.

My work in photonics has focussed on computational modelling of various aspects of photonic crystals - composite materials which exhibit a periodic variation in their refractive index, leading to novel and interesting optical properties. An example of some results of this work may be seen in the paper Effects of disorder in two-dimensional photonic crystal waveguides.

**Can supervise:**Yes

## Book Chapters

Botten, L.C., McPhedran, R.C., de Sterke, C.M., Nicorovici, N.A., Asatryan, A.A., Smith, G.H., Langtry, T.N., White, T., Fussell, D.P. & Kuhlmey, B. 2006, 'From Multipole Methods to Photonic Crystal Device modeling' in Yasumoto K (ed),

View/Download from: UTSePress |

*Electromagnetic Theory and Applications for Photonic Crystals*, Taylor & Francis, Florida, USA, pp. 47-122.View/Download from: UTSePress |

NA

Botten, L.C., McPhedran, R.C., Nicorovici, N.A., Asatryan, A.A., de Sterke, C.M., Robinson, P.A., Busch, K., Smith, G.B. & Langtry, T.N. 2003, 'Rayleigh multipole methods for photonic crystal calculations' in Priou A; Itoh T (eds),

*Electromagnetic applications of photonic band gap materials and structures*, Massachusetts Institute of Technology, USA, pp. 21-60.## Conference Papers

Zinder, Y., Nicorovici, N.A. & Langtry, T.N. 2011, 'Mathematica based platform for self-paced learning', ICTMT10, Portsmouth, United Kingdom, July 2011 in

*Proceedings of the 10th International Conference on Technology in Mathematics Teaching (ICTMT10)*, ed Marie Joubert; Alison Clark-Wilson; Michael McCabe, University of Portsmouth, United Kingdom, pp. 203-208. One of the major challenges in teaching applied mathematics is the large amount of calculation involved in many practical mathematical methods. On one hand, mastering these methods requires students to gain experience in performing all steps of a calculation. This experience is crucial for gaining an understanding of the methods, their capabilities and limitations, and cannot be replaced by black box type commercial software which simply displays the results of calculations and gives no insight into the nature of the operations performed. On the other hand, the calculations involved in many modern mathematical methods are tedious and time consuming with fatal results caused by even minor computational mistakes. The advent of computer algebra systems such as Mathematica opened new horizons in teaching mathematics. The ability to perform symbolic calculations in combination with powerful graphics and programming capabilities makes it possible to develop software that provides students with the opportunity for step by step exploration of mathematical procedures. We present the results of an ongoing research project aimed at the development of a Mathematica based platform which allows academics to develop software for self-paced learning.

Zinder, Y., Nicorovici, N.A. & Langtry, T.N. 2010, 'Mathematica based platform for self-paced learning', EDULEARN, Spain, July 2011 in

*Proceedings of EDULEARN 10*, ed Standl, B., International Association of Technology, Education and Development, Spain, pp. 323-330. One of the major challenges in teaching applied mathematics is the large amount of calculation involved in many practical mathematical methods. On one hand, mastering these methods requires students to gain experience in performing all steps of a calculation. This experience is crucial for gaining an understanding of the methods, their capabilities and limitations, and cannot be replaced by black box type commercial software which simply displays the results of calculations and gives no insight into the nature of the operations performed. On the other hand, the calculations involved in many modern mathematical methods are tedious and time consuming with fatal results caused by even minor computational mistakes. The advent of computer algebra systems such as Mathematica opened new horizons in teaching mathematics. The ability to perform symbolic calculations in combination with powerful graphics and programming capabilities makes it possible to develop software that provides students with the opportunity for step by step exploration of mathematical procedures. We present the results of an ongoing research project aimed at the development of a Mathematica based platform which allows academics to develop software for self-paced learning.

Maher, A., Lake, M., Langtry, T.N. & Hill, C. 2007, 'A proteomics laboratory system and digital mentor', Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Perth, Australia, October 2007 in

View/Download from: UTSePress |

*APAC07*, ed Appelbe, W, Australian Partnership for Advanced Computing, Canberra, Australia, pp. 1-19.View/Download from: UTSePress |

The typical small biological laboratory is cmposed of electronic scientific equipment and wet laboratory equipment where staff record reesults into paper workbooks. There is oftern a lack of expert staff to guide new users, and hard won experience is locked away with individual staff and their workbooks. The Proteomics Laboratory System and Digital Mentor aims to: facilitate the capture, analysis and sharing of results; facilitate the creation and documentation of standardised work flows and experimental procedures; and mentore inexperienced users in best practice when engaged in particular procedures in the laboratory. In this paper we discuss the design of the system and report on the initial production release. This is nased on a relational database and web-server framework. A novel aspect of the system is the exploitation of custom extensions to a wiki-style interface that add functionality for non-expert end-users.

Langtry, T.N., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2004, 'Localisation and disorder in the design of 2D photonic crystal devices', Computational Techniques and Applications Conference, Sydney, Australia, July 2003 in

View/Download from: Publisher's site

*ANZIAM Journal - 11th Biennial Computational Techniques and Applications Conference: CTAC-2003*, ed NA, Cambridge University Press, Australia, pp. 744-758.View/Download from: Publisher's site

Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Asatryan, A.A., Botten, L.C., McPhedran, R.C., de Sterke, C.M., Langtry, T.N. & Nicorovici, N.A. 2004, 'Conductance of photons and Anderson localization of light', Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies, San Francisco, USA, May 2004 in

View/Download from: UTSePress

*Conference on Lasers and Electro-Optics/International Quantum Electronics Conference and Photonic Applications Systems Technologies*, ed NA, Optical Society of America, USA, pp. 860-861.View/Download from: UTSePress

Conductance properties of photons in disordered two-dimensional photonic crystals is calculated using exact multipole expansions technique. The Landauer+s two-terminal formula is used to calculate average of the conductance, its variance and the probability density distribution.

Asatryan, A.A., Botten, L.C., White, T.P., de Sterke, C.M., McPhedran, R.C. & Langtry, T.N. 2003, 'Modelling of complex waveguides structures embedded in photonic crystals', Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, Melbourne, Australia, July 2003 in

View/Download from: UTSePress

*COIN/ACOFT 2003. Proceedings on the Optical Internet and Optical Fibre Technology*, ed Nirmalathas, TA; Park J, COIN/ACOFT 2003, Melbourne, Australia, pp. 145-148.View/Download from: UTSePress

de Sterke, C.M., Botten, L.C., White, T.P., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2003, 'Photonic bandgap effects as a basis for novel compact devices', Conference on the Optical Internet / Australian Conference on Optical Fibre Technology 2003, Melbourne, Australia, July 2003 in

View/Download from: UTSePress

*COIN/ACOFT 2003. Conference Proceedings on the Optical Internet and Optical Fibre Technology*, ed Nirmalatha TA; Park J, COIN/ACOFT 2003, Melbourne, Australia, pp. 133-136.View/Download from: UTSePress

Langtry, T.N., Botten, L.C., Asatryan, A.A., Byrne, M.A., Bourgeois, A. & McPhedran, R.C. 2003, 'Computational modelling of photonic crystals', Conference and Exhibition on Advanced Computing, Grid Application and eResearch, Gold Coast, Australia, September 2003 in

View/Download from: UTSePress

*Proceedings of the APAC Conference and Exhibition on Advanced Computing, Grid Application and eResearch*, ed Young J; Chhabra R, Australian Partnership for Advanced Computing (APAC, Gold Coast, QLD, pp. 1-19.View/Download from: UTSePress

Botten, L.C., Asatryan, A.A., Langtry, T.N., de Sterke, C.M. & McPhedran, R.C. 2002, 'Propagation of Photonic Crystal Waveguides', Congress of the Australian Institute of Physics, Sydney, Australia, July 2002 in

View/Download from: UTSePress

*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, ed Nielsen D, Australian Institute of Physics, Sydney, Australia, pp. 289-291.View/Download from: UTSePress

Langtry, T.N., Botten, L.C., de Sterke, C.M., Asatryan, A.A. & McPhedran, R.C. 2002, 'Effects of Disorder in Photonic Crystal Waveguides', Congress of the Australian Institute of Physics, Sydney, Australia, July 2002 in

View/Download from: UTSePress

*Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002.*, ed Nielsen D, Australian Institute of Physics, Sydney, Australia, pp. 310-312.View/Download from: UTSePress

Langtry, T.N., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2001, 'Local Density of States Calculations for Photonic Crystals', International Conference and Exhibition on High Performance Computing in the Asia-Pacific Region, Gold Coast, September 2001 in

View/Download from: UTSePress

*Proceedings of HPC Asia 2001*, ed Young J, Queensland Parallel Supercomputing Facilty and Australian Partnership for Advasnced Computing, Brisbane, pp. 0-0.View/Download from: UTSePress

## Journal Articles

Langtry, T.N., Giokaris, P., Milthorpe, B.K. & Lord, M.S. 2013, 'Parameter estimation for a model of fibronectin adsorption onto hydroxylapatite, oxidised polystyrene and nanostructured silica',

View/Download from: UTSePress |

*ANZIAM*, vol. 54, no. CTAC2012, pp. C429-C445.View/Download from: UTSePress |

Fibronectin is a protein present in blood and the extracellular matrix which has important roles in cell adhesion and migration, wound healing and blood clotting. Three models of fibronectin adsorption, on various substrates of interest to biochemists, are compared. The first model (of Langmuir) is expressed explicitly as a time dependent function for the mass of protein adsorbed. The second model is a modification of the scaled particle theory of Reiss et al. [J. Chem. Phys., 31:369--380, 1959] and takes into account the probability of a molecule finding a sufficiently large vacant area on the adsorbing substrate surface. The third model extends the second model to the case in which molecules may expand the radius of their contact with the substrate upon adsorption. We used datasets obtained from experiments to compare the models. The Langmuir model is straightforward to fit to a dataset. The remaining models are fitted using a steepest descent method to minimise least squares error. We describe initial estimates for parameters for this procedure and compare the quality of fit of the models.

Chew, K., Langtry, T.N., Zinder, Y., Yu, Q. & Li, L. 2012, 'Estimation of biochemical parameters from leaf photosynthesis',

View/Download from: UTSePress

*ANZIAM Journal*, vol. 53, pp. C218-C235.View/Download from: UTSePress

The objective of measuring leaf photosynthesis using infrared gas analysis is to determine key indicators of plant eco-physiology, including light and CO2 compensation and saturation points, and critical thresholds of temperature. These and other biochemical parameters in photosynthesis models define specific response curves of photosynthetic rate to environmental variables, such as light intensity, temperature, and CO2. Since these parameters cannot regularly be measured in the field, modellers normally adopt laboratory values as universal ones even though the values of these parameters may vary across plant species. This study investigates the identification of parameter values from data sets obtained from field measurement.

Jay, K., Chaumet, P.C., Langtry, T.N. & Rahmani, A. 2010, 'Optical binding of electrically small magnetodielectric particles',

View/Download from: UTSePress | Publisher's site

*Journal of Nanophotonics*, vol. 4, pp. 1-9.View/Download from: UTSePress | Publisher's site

An ensemble of spherical particles with arbitrary dielectric permittivity and magnetic penneability was considered in the dipole approximation. Each particle was described by complex electric and magnetic polarizabilities. A computational approach based on the coupled dipole method, also called the discrete dipole approximation, was used to derive the optical force experienced by each particle due to an incident electromagnetiG..Ji.eld and the fields scattered by all other particles. This approach is general and can handle material dispersion and losses. In order to illustrate this approach, we studied the case of two spherical particles separated by a distance d, and illuminated by an incident plane wave whose wave vector is normal to the axis of the particles. We computed the optical force experienced by each particle in the direction of the beam (radiation pressure), and perpendicular to the beam (optical binding) for particles with positive and negative refractive indices. We also considered the effect of material losses.

Asatryan, A.A., Botten, L.C., Byrne, M.A., Langtry, T.N., Nicorovici, N.A., McPhedran, R.C., de Sterke, C.M. & Robinson, P.A. 2005, 'Conductance of photons in disordered photonic crystals',

View/Download from: UTSePress

*Physical Review E*, vol. 71, no. 3, pp. 036623-1-036623-8.View/Download from: UTSePress

The conductance of photons in two-dimensional disordered photonic crystals is calculated using an exact multipole-plane wave method that includes all multiple scattering processes. Conductance fluctuations, the universal nature of which has been establis

Botten, L.C., White, T.P., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Photonic crystal devices modelled as grating stacks: matrix generalisations of thin film optics',

View/Download from: UTSePress | Publisher's site

*Optics Express*, vol. 12, no. 8, pp. 1592-1604.View/Download from: UTSePress | Publisher's site

A rigorous semi-analytic approach to the modelling of coupling guiding and propagation in complex microstructures embedded in two-dimensional photonic crystals is presented. The method, which is based on Bloch mode expansions an generalised Fresnel coefficients is shown to be able to treat photonic crystal devices in ways which are analogous to those used in thin film optics with uniform media. Asymptotic methods are developed and exemplified through the study of a serpentine waveguide, a potential slow wave device.

White, T.P., Botten, L.C., de Sterke, C.M., McPhedran, R.C., Asatryan, A.A. & Langtry, T.N. 2004, 'Bloch mode scattering matrix methods for modelling extended photonic crystal structures. II: Applications',

View/Download from: UTSePress | Publisher's site

*Physical Review E*, vol. 70, no. 5, pp. 1-10.View/Download from: UTSePress | Publisher's site

The Bloch mode scattering matrix method is applied to several photonic crystal waveguide structures and devices, including waveguide dislocations, a Fabry-Perot resonator, a folded directional coupler, and a Y-junction design. The method is an efficient tool for calcuating the properties of extended photonic crystal (PC) devices, in particular whenthe device consists of a small number of distict photonic crystal structures, or for long propagation lengths through uniform PC waveguides. The physical insight provided by the method i sused to derive simple, semianalytic models that allow fast and efficient calculations of complex photonic crystal structures. We discuss the situations in which such simplifications can be made and provide examples.

Langtry, T.N., Botten, L.C., Asatryan, A.A., Byrne, M.A. & Bourgeois, A. 2004, 'Localisation and disorder in the design of 2D photonic crystal devices',

View/Download from: UTSePress |

*ANZIAM Journal*, vol. 45, pp. 744-758.View/Download from: UTSePress |

Photonic crystals are meta-materials that can inhibit the propagation of light in all directions for specific wavelength ranges. Material or structural defects can be introduced into the crystal to cause localised modes, providing the ability to mould the flow of light on the wavelength scale and allowing the development of miniaturised, integrated photonic devices. For this reason, photonic crystals will likely be key building blocks for future micro-optical and communication technology. In this paper, we examine the Bloch mode modelling of 2D photonic crystal structures with application to the analysis of photonic crystal waveguides and their susceptibility to disorder, which provides a framework for studying fabrication tolerances in realistic devices.

Botten, L.C., White, T.P., Asatryan, A.A., Langtry, T.N., de Sterke, C.M. & McPhedran, R.C. 2004, 'Bloch Mode Scattering Matrix Methods For Modeling Extended Photonic Crystal Structures. I. Theory',

View/Download from: UTSePress

*Physical Review E*, vol. 70, no. 5, pp. 1-13.View/Download from: UTSePress

We present a rigorous Bloch mode scattering matrix method for modeling two-dimensional photonic crystal structures and discuss the formal properties of the formulation. Reciprocity and energy conservation considerations lead to modal orthogonality relati

Asatryan, A.A., Robinson, P.A., McPhedran, R.C., Botten, L.C., de Sterke, C.M., Langtry, T.N. & Nicorovici, N.A. 2003, 'Diffusion and anomalous diffusion of light in two-dimensional photonic crystals',

View/Download from: UTSePress

*Physical Review E*, vol. 67, no. 3, pp. 1-8.View/Download from: UTSePress

Langtry, T.N., Asatryan, A.A., Botten, L.C., de Sterke, C.M., McPhedran, R.C. & Robinson, P.A. 2003, 'Effects of disorder in two-dimensional photonic crystal waveguides',

View/Download from: UTSePress

*Physical Review E*, vol. 68, no. 2, pp. 026611-1-026611-11.View/Download from: UTSePress

Langtry, T.N., Botten, L.C., Asatryan, A.A. & McPhedran, R.C. 2003, 'Monte Carlo modelling of imperfections in two-dimensional photonic crystals',

View/Download from: UTSePress | Publisher's site

*Mathematics And Computers In Simulation*, vol. 62, no. 3-6, pp. 385-393.View/Download from: UTSePress | Publisher's site

In this paper we describe a Monte Carlo simulation of imperfections in photonic crystals, a new class of materials with optical properties that offer promise in a range of potential applications in the areas of information and communications technology. We describe the relevant physical and structural properties of these materials and outline the derivation of a theoretical model. We then present a Monte Carlo investigation of the tolerance of these materials to fabrication defects.

Botten, L.C., Asatryan, A.A., Langtry, T.N., White, T.P., de Sterke, C.M. & McPhedran, R.C. 2003, 'Semianalytic treatment for propagation in finite photonic crystal waveguides',

View/Download from: UTSePress | Publisher's site

*Optics Letters*, vol. 28, no. 10, pp. 854-856.View/Download from: UTSePress | Publisher's site

e present a semianalytic theory for the properties of two-dimensional photonic crystal waveguides of finite length. For single-mode guides, the transmission spectrum and field intensity can be accurately described by a simple two-parameter model. Analogies are drawn with Fabry+Perot interferometers, and generalized Fresnel coefficients for the interfaces are calculated.

Langtry, T.N., Coupland, M.P. & Moore, B.J. 2003, 'Mathematica in context',

View/Download from: UTSePress | Publisher's site

*International Journal of Mathematical Education in Science & Technology*, vol. 34, no. 5, pp. 699-718.View/Download from: UTSePress | Publisher's site

Thornton, B.S., Nguyen, H.T., Hung, W., Hirst, C., Thornton-Benko, E. & Langtry, T.N. 2001, 'Breast Screening Outcomes: Communications Problems, Chaos Relationship and Control Theory',

*Canadian Applied Mathematics Quarterly*, vol. 9, no. 4, pp. 377-401. Langtry, T.N. 2001, 'A construction of higher-rank lattice rules',

View/Download from: UTSePress | Publisher's site

*Mathematics and Computers in Simulation*, vol. 55, no. 1-3, pp. 103-111.View/Download from: UTSePress | Publisher's site

Lattice rules are quasi-Monte Carlo methods for numerical multiple integration that are based on the selection of an s-dimensional integration lattice. The abscissa set is the intersection of the integration lattice with the unit hypercube. It is well-known that the abscissa set of a lattice rule can be generated by a number of fixed rational vectors. In general, different sets of generators produce different integration lattices and rules, and a given rule has many different generator sets. The rank of the rule is the minimum number of generators required to span the abscissa set. A lattice rule is usually specified by a generator set, and the quality of the rule varies with the choice of generator set. This paper describes a new method for the construction of generator sets for higher-rank rules that is based on techniques arising from the theory of simultaneous Diophantine approximation. The method extends techniques currently applied in the rank 1 case.

Langtry, T.N. 1999, 'Lattice rules of minimal and maximal rank with good figures of merit',

View/Download from: UTSePress | Publisher's site

*Journal of Computational and Applied Mathematics*, vol. 112, no. 1-2, pp. 147-164.View/Download from: UTSePress | Publisher's site

For periodic integrands with unit period in each variable, certain error bounds for lattice rules are conveniently characterised by the figure of merit rho, which was originally introduced in the context of number theoretic rules. The problem of finding good rules of order N (that is, having N distinct nodes) then becomes that of finding rules with large values of rho. This paper presents efficient search methods for the discovery of rank 1 rules, and of maximal rank rules of high order, which possess good figures of merit.

Langtry, T.N. 1996, 'An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules',

View/Download from: UTSePress | Publisher's site

*Mathematics Of Computation*, vol. 65, no. 216, pp. 1635-1662.View/Download from: UTSePress | Publisher's site

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in R(s) and are generalizations of 'number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)-themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit rho, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order N (that is, having N nodes) then becomes that of finding rules with large values of rho. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

Langtry, T.N. 1995, 'The determination of canonical forms for lattice quadrature rules',

View/Download from: Publisher's site

*Journal of Computational and Applied Mathematics*, vol. 59, no. 2, pp. 129-143.View/Download from: Publisher's site

Lattice rules are equal weight numerical quadrature rules for the integration of periodic functions over the s-dimensional unit hypercube U-s = [0, 1)(s). For a given lattice rule, say Q(L), a set of points L (the integration lattice), regularly spaced in all of R(s), is generated by a finite number of rational vectors. The abscissa set for Q(L) is then P(Q(L))=L boolean AND U-s. It is known that P(Q(L)) is a finite Abelian group under addition module the integer lattice Z(s), and that QL(f) may be written in the form of a nonrepetitive multiple sum, known as a canonical form, in which + denotes addition module Z(s). In this form, z(i) is an element of Z(S), m is called the rank and n(1), n(2),..., n(m) are called the invariants of Q(L), and n(i+1)\n(i) for i = 1,2,..., m-1 . The rank and invariants are uniquely determined for a given lattice rule. In this paper we provide a construction of a canonical form for a lattice rule Q(L), given a generator set for the lattice L. We then show how the rank and invariants of QL may be determined directly from the generators of the dual lattice L perpendicular to.

Thornton, B.S. & Langtry, T.N. 1988, 'Prescheduling graphic displays for optimal cancer therapies to reveal possible tumour regression or stabilisation',

View/Download from: UTSePress

*Journal of Medical Systems*, vol. 12, no. 1, pp. 31-41.View/Download from: UTSePress

The paper describes an adaptive control approach to the problem of the treatment of solid tumors. The evolution with time t of the state of a tumor is modelled by a two-compartment system, governed by two differential equations forming an autonomous system under therapy control u, dy1/dt = f1(y1,y2;u) dy2/dt = f2(y1,y2;u), where y1 and y2 are the number of proliferating and nonproliferating cells, respectively. The output is analyzed in the phase plane y1y2. The control problem is that of restricting the tumor state to a predetermined region of the plane by selecting a suitable change in therapy control u, e.g., modality and dosage, when the state solution intersects the boundary of this region and the ratio y1/y2 of proliferating to nonproliferating cells is displayed together with an elapsed time scale. Then, consequent selection of a suitable therapeutic sequence may be assisted by the use of a data base as part of an expert system. The process is repeated at each intersection of the prescribed boundary. Such sequences may lead to stabilization of the system through the appearance on a computer display screen of a stable equilibrium point or a limit cycle.