How do microbes evolve antibiotic resistance so rapidly? Can we use insights gained from such evolutionary processes to design "intelligent" robots that can adapt to unforeseen situations with novel solutions? How do thoughts, memories, and consciousness emerge from interactions of billions of neurons in our brains? Can we borrow ideas from the exquisite workings of the brain to create computer programs that can recognize complex patterns? How do different combinations of genes affect a person's likelihood of getting heart disease? What types of interventions are likely to be effective in limiting the spread of a global flu pandemic? How can we redesign our transportation systems to promote economic growth and connect people to one another, while avoiding further environmental degradation, congestion, and urban sprawl? How can we better predict the paths and strengths of hurricanes? Can we restore health and stability to our earth's ecology and environment?
Seemingly disparate questions such as the ones posed above all share the fact that they deal with systems in which, as aptly stated by Aristotle (384-322 BC), "the whole is more than the sum of the parts." In other words, these systems are non-additive ("nonlinear"). As scientists and engineers have become increasingly aware of the interconnectedness of essentially all aspects of the world in which we live, there has been growing interest in developing effective approaches to studying these so-called "complex systems."
Complex systems comprise sets of many adapting entities that interact with each other and with their environment in nonlinear, dynamic ways. Interactions between entities at a given scale give rise to emergent properties at larger scales in space and/or time through self-organization, without any global knowledge or central control.
The behaviors of complex systems are difficult to predict, in part because the outcomes may be very sensitive to small changes in system inputs, and because they often contain feedback loops, where system outputs feed back into the system as inputs and can either ramp up or damp down system processes. Such dynamic behaviors between entities at a given scale can often result in semi-stable patterns in space and/or time at a higher scale.
Complex systems thus exhibit qualitatively distinct properties at different scales, and can be viewed as systems of systems. For example, atoms can interact to form molecules, molecules can interact to form cells, cells can interact to form organs, organs can interact to form organisms, organisms can interact with each other and their environment to form societies and ecosystems, and so on. Defining the scales and boundaries of a complex system of interest is thus necessarily arbitrarily, depending on the researchers' perspective and the questions they're asking. These simplifications introduce further uncertainty into any system predictions.
Nonetheless, despite these difficulties and the vast ranges in scales of complex systems of interest, several mathematical and computational tools are arising that enable scientists and engineers to gain insight into how these systems work, and how to engineer or interface with them in desirable ways.
Emergent system behaviors cannot be predicted from a reductionist, descriptive approach to studying the entities alone, so the study of complex systems must also include an understanding of the nonlinear interactions between these entities. Consequently, complex systems are "computationally irreducible," and the behaviors of interacting entities must be modeled in order to observe and study how phenomena emerge at larger scales. Similarly, making sense of observed system properties requires sophisticated nonlinear spatio-temporal analysis.
Core theory and methods for complex systems research (that transcend a variety of applications and a variety of scales) therefore include agent-based modeling (including cellular automata, real-space models, network-based models), computational evolution and evolutionary computation, study of structure and dynamics of complex networks, artificial neural networks, scaling phenomena and laws, stochastic and Markovian processes, nonlinear differential equations, chaos theory, fractals, numerical optimization, multi-scale modeling, nonlinear spatio-temportal statistical pattern recognition, nonlinear data mining, and distributed sensing and control. UVM has computer scientists, mathematicians, statisticians and engineers with complementary expertise in all of these areas. These faculty offer core complex systems curricula and collaborate with researchers across the UVM campus and outside UVM, in the study of complex systems in a variety of application domains including various fields of biology, ecology, environmental science, social psychology, computing, governance, economics, transportation, medicine, etc.
The Vermont Complex Systems Center serves as a hub for the unifying theme of complex systems activities across the UVM campus, addressing some of the world's most pressing problems, such as remediating major environmental concerns, creating sustainable transportation systems, engineering adaptive distributed systems, maintaining species diversity, unraveling the mysteries of the genetic code, and informing public policy.
There are many excellent descriptions of complex systems available on the web and elsewhere. The Wikipedia page on Complex Systems is a great place to start, for more details and additional links to other complex systems sites.