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Bifurcations in one-dimensional piecewise smooth continuous maps by Soumitro Banerjee

701 views · Published 26 November 2018 · 1:03:14 · Indexed 28 September 2026

Channel: International Centre for Theoretical Sciences · 2018 · Science & Technology

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PROGRAM

DYNAMICS OF COMPLEX SYSTEMS 2018

ORGANIZERS

Amit Apte, Soumitro Banerjee, Pranay Goel, Partha Guha, Neelima Gupte, Govindan Rangarajan and Somdatta Sinha

DATE: 16 June 2018 to 30 June 2018

VENUE: Ramanujan hall for Summer School held from 16 - 25 June, 2018; Madhava hall for Workshop on Complex Networks held from 26 - 30 June, 2018

This Summer Program on Dynamics of Complex Systems is the third in the series. The theme for the program this year is non-smooth dynamical systems and complex networks.

The study of bifurcations, i.e., qualitative change in the behavior of a dynamical system as a set of parameters of the system is varied, is an important and well-studied area of research for such systems.  A large fraction of the dynamical systems theory is devoted to the study of smooth (continuously differentiable) dynamical systems. However, there are also systems encountered commonly in science and engineering, where a dynamical system is given by two or more different sets of differential equations, each defined for a compartment of the phase space. As the state moves from one compartment to another, the equations defining the evolution change. Examples of such systems, where continuous-time evolution is punctuated by discrete switching events, are:

Power Electronic Circuits
Systems involving relays
Mechanical systems with impacts or stick-slip motion
Nonlinear circuits like Colpitt's oscillator, Chua's circuit etc.
Continuous systems with phased operation (bouncing ball, walking robots, etc.)
Continuous systems controlled by discrete logic (aircraft autopilot modes, thermostats, chemical plants with valves and pumps, automobile automatic transmissions).
The study of bifurcations in such non-smooth dynamical systems is an active and emerging research field. Such systems exhibit a new dynamical phenomenon known as border collision bifurcation which may result in robust chaos, multiple attractor bifurcations, dangerous bifurcations, etc. In understanding these phenomena and to analyse the bifurcations observed in practical systems, one needs to know under what conditions a border collision will result in a particular type of observable system behavior (say, a periodic orbit bifurcating directly into chaos).

In the School (June 16 to 25, 2018), the basic theory of non-smooth dynamical systems will be taught, and the students will be exposed to the tools and techniques used in investigations in this area. Speakers in the School will include:

Soumitro Banerjee, IISER Kolkata
David Simpson, Massey University, New Zealand
Paul Glendin ning, University of Manchester, UK
Viktor Avrutin, University of Stuttgart, Germany
To apply for the school, please see the following webpage. It is the applicant's responsibility to make sure that the letters are received on time. Selected applicants will be informed by Saturday 31 March 2018.

An equally important and emerging theme of research in dynamical systems is the study of complex networks, which are ubiquitous in nature (metabolic networks, ecological networks, neural networks, river networks), social and economic systems (facebook, banking networks, financial networks), as well as in technology (power grids, internet). The importance of this field was recognized almost immediately by the Indian dynamical systems researchers, and thus there is a community, spread all over the country and also across disciplines, which is eager to pick up new directions and technically equipped to contribute usefully in this field. In order to strengthen further research in this area, the school will have a set of four lectures on elementary topics in networks, so that students acquire some basic information and techniques about networks.

This will lead to the Workshop on Complex Networks (WCN) from June 26 to 30, 2018. There will be about 20 researchers in the field, who would present their own work, and also outline problems where collaborative work can potentially start. Workshop topics include:

Networks: Structure and Function
Network characterisers: spectral and topological
Traffic on networks
Time series networks and  inversion
Applications to real world phenomena.
 
CONTACT US: [email protected]
PROGRAM LINK: https://www.icts.res.in/program/dcs2018

Table of Contents (powered by https://videoken.com)
0:03:10 The border collision normal form
0:05:54 The border collision bifurcations
0:12:38 Border crossing bifurcations
0:14:23 Case 6: "Period-1 - period-2"
0:17:57 Case 7: "Period-1 - no attractor"
0:19:42 Case 8: "Period-1 - high periodic or chaotic attractor"
0:24:06 The parameter space
0:27:42 Border-Collision Bifurcations in One-Dimensional Discontinuous Maps
0:38:27 Case 1: Similarly, the three points in an LLR orbit are
0:40:18 Case 2: For negative discontinuity,
0:43:49 Case 4: For positive
0:46:07 Case 5: 
0:47:34 Case 6:
0:49:28 Case 7:
0:50:41 Case 8:
0:53:43 Case 9:
0:55:46 Thank You

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