1. Lecture note for 59-545/445

2. What is a dynamical system? In an informal way, one can think of a dynamical system as being either of two things: - A MATHEMATICAL model that describes the evolution with respect to time of some quantity, e.g., the simple equation dx/dt = - x is a simple dynamical system that tells us how the quantity x(t) changes with time (it decays exponentially). - A PHYSICAL system that has observable properties that can evolve in time, e.g., the position theta(t) of a mechanical pendulum, the voltage V(t) of an electronic circuit, or the x-velocity Vx(t,x,y,z) at a particular point (x,y,z) in space at time t of a fluid flowing in a pipe.

3. What is nonlinear dynamics the rule used to update the state of the system is nonlinear. In a map of the form: x[n+1] = f( x[n] ), where f(x) is a nonlinear function. In chemical reaction rate laws: d[A]/dt = k*[B]*[C] 2

4. Nonlinear Chemical Dynamics Plot the rate law as a function of the concentration of reactants for different reaction schemes. In closed systems the only concentrations which vary in an oscillatory way are those of the intermediates. There is generally a monotonic decrease in reactant concentrations and a monotonic, but not necessarily smooth, increase in those of the products. The free energy even of oscillatory systems decreases continuously during the course of reaction: ΔG does not oscillate.

5. Feedbacks in chemical reactions A vital constituent of any chemical reactions that is going to show oscillations or other bifurcations is that of “feedback”. Isothermal feedbacks Autocatalysis Autoinhibition Non-isothermal self-heating process (Arrhenius law: k(T) = A exp(-E/RT).

6. Chemical feedback Example of chemical feedback by chain branching H + O 2 → OH + O (k1) OH + H 2 → H 2 O + H O + H 2 → OH + H Overall: H + 3H 2 + O 2 → 3H + 2H 2 O Assuming the first step is a rate determining step: Then: d[H]/dt = 2k1[H][O 2 ] (show in class)

7. Chemical feedback Example of chemical feedback by autocatalysis HBrO 2 + BrO 3 - BrO 3 - + H + →2BrO 2. + H 2 O BrO 2. + Ce 3+ + H + → HBrO 2 + Ce 4+ Overall HBrO 2 + BrO 3 - + 3H + + 2Ce 3+ → 2HBrO 2 + 2Ce 4+ + H 2 O With a rate law: [HBrO 2 ][ /dt = + k [HBrO 2 ][HBrO 2 ][H + ]

8. Consequence of nonlinear feedbacks A + 2B → 3B; operated in a CSTR (thermodynamic branch and flow branch). Oscillations Multistability Hysteresis (Extinction: jump to the flow branch; ignition: jump to the equilibrium branch). Analogous to the double-well potential.

9. Modes of representing data Time series Constraint-response plot Phase diagrams Phase portraits

10. IMPLICATIONS OF NONLINEAR DYNAMICS FOR DIFFERENT DISCIPLINES PHYSICS: - Nonlinear dynamics (ND) is a natural continuation of classical mechanics, e.g., the behavior of planets under gravity, charges in a plasma, mechanical systems with many parts. - ND is also a natural continuation of statistical mechanics, the physics of behaviors that arise from many simple components that interact with each other. There is an especially close intellectual and historical link between the theory of phase transitions and several areas of nonlinear dynamics. - Nonlinear dynamics represents a major new young area of physics research: sustained nonequilibrium physics, for which no deep fundamental theory yet exists of a power comparable to thermodynamics or statistical physics. Behavior of nonequilibrium systems is profoundly different from solids, gases, liquids in equilibrium and new concepts are needed to understand this behavior.

11. IMPLICATIONS OF NONLINEAR DYNAMICS FOR DIFFERENT DISCIPLINES ENGINEERING: -Engineering traditionally concerns the design and control of systems. Very difficult to achieve these goals without a basic understanding of what is going on: -Architecture: Tacoma Narrows bridge near Seattle, see the movie http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html -Biomedical engineering: fibrillation of the heart: how to stop or control? Similarly for epilepsy of the brain: why does it commence, how to stop it? -Powerful lasers: single large laser, grid of small lasers, either case you are in trouble. - Fluctuations in the American power grid. -Fusion plasma reactors: main problem is instability. -These systems all have in common a complex evolution in both time and space. -There has been some significant useful progress in how to control complex systems, and this progress was made by leading members of the nonlinear community.

12. Implications Of Nonlinear Dynamics for Biological media