2005 E arth S cience S ummer A cademy

1. 2005Earth Science Summer Academy Virginia Department of Education Lynn S. Fichter • Department of Geology/Environmental Science • E-mail: fichtels@jmu.edu • James Madison University • http://csmres.jmu.edu/geollab/Fichter/Fichter/Fichterls.html • June, 2005 • http://www.jmu.edu/geology/evolutionarysystems/Academy2005.shtml

2. Hysteresis a.k.a. Bistable Behavior OR The lag in response between a cause and its effect. OR The failure of a property that has been changed by an external agent to return to its original value when the cause of the change is removed. OR The ability of a system to exist in different states under the same conditions.

3. Hysteresis Or, when a system has more than one path available to it, and periodically switches between them, when and how does it choose between them? As in the logistic system

4. Hysteresis Or, when a system has more than one path available to it, and periodically switches between them, when and how does it choose between them? As in the Lorenz attractor system Run Lorenz Attractor Lorenz Applet http://www.csounds.com/mastering/em_10.html http://www.cmp.caltech.edu/~mcc/chaos_new/Lorenz.html

5. Hysteresis Or, when a system has more than one path available to it, and periodically switches between them, when and how does it choose between them? As in an oscillating Reaction-Diffusion system Color A Reaction Moving Waves Color B Inhibition http://delfin.klte.hu/~gasparv/menuh.html http://people.musc.edu/~alievr/rubin.html Note that all these systems are far from equilibrium; hysteresis is only possible in actively open systems dissipating lots of energy.

6. Hysteresis All complex systems exhibit this behavior. • We know this fact by experience in our own lives, but the phenomena has not been systematically taught in the science most of us have had. • Therefore, we are not consciously aware of it. 1. Do not recognize it for what it is when we see it. 2. Do not have a theoretical framework to fit it in. 3. Find it hard to anticipate in real time that a system we are experiencing is going to undergo a dramatic shift . . . perhaps soon. • We do not recognize the symptoms.

7. Hysteresis It is also a bias toward gradualism – the idea that all changes are slow and gradual, requiring long periods of time. • Allows us to argue, for example, that global warming is not an issue since any changes will require thousands of years. • Or at least it will not happen in our life time. • And biological evolution is a slow process. Maxim for the Day You can only see what you are looking for . . . And, you can’t see what you are not looking for . . .

8. Hysteresis Description One The Lag Between Cause and Effect The sand pile builds . . . grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . by grain . . . Description One until it reaches the critical state . . . and avalanches building building building avalanche avalanche avalanche Thus we see there is a lag between cause (accumulation of individual sand grains), and the effect (avalanche).

9. Hysteresis Description One The Lag Between Cause and Effect Fractal sand supply Now, imagine the sand supply follows a power law (or is fractal), with different numbers of grains falling at different times. Avalanches will also be fractal, and follow a power law distribution. A time series plot like this graphs bistable behavior (hysteresis) at many scales of observation. Notice the ups and downs are not symmetrical, meaning rises and falls are not symmetrical. There is a lag between cause and effect

10. Hysteresis Description One The Lag Between Cause and Effect The Tipping Point The prevalence of this phenomena of lags between cause and effect was explored by Malcolm Gladwell in “The Tipping Point” "The best way to understand the dramatic transformation of unknown books into bestsellers, or the rise of teenage smoking, or the phenomena of word of mouth or any number of the other mysterious changes that mark everyday life, is to think of them as epidemics. Ideas and products and messages and behaviors spread just like viruses do." Little changes can have big effects; when small numbers of people start behaving differently, that behavior can ripple outward until a critical mass or "tipping point" is reached, changing the world. Return to Systems

11. Hysteresis as Bistable Behavior Description Two The System can exist in distinctly different states under the same conditions. Over shoots left Over shoots right Over shoots left Over shoots right Description Two X = dependent variable Time Series Hysteresis Diagram r = independent variable

12. Hysteresis as Bistable Behavior Description Two The System can exist in distinctly different states under the same conditions. First, a little slight of hand . . . State 1 Time Series is now the Bistable Variable Time Series at one “r” value “r” is still the external energy driving the system State 2 X as Dependent Variable X is now the Driving Variable Bistable behavior requires that two variables be coupled in a positive/negative feedback loop. • A rise in one variable causes the other variable to fall, and vice versa. • Cannot speak of independent and dependent variables since they are coupled.

13. Mechanisms of Bistable Behavior Cannot speak of independent and dependent variables since variables are coupled. There are other external variables or energy sources driving or responding to the system. State 1 Low Path Dynamics What ever is driving the Driving Variable to the left . . . Bistable Variab le . . . The more it moves to the left the more the Bistable Variable becomes unstable in the lower position. State 2 At some critical (sensitive dependent) point the Bistable Variable shifts (avalanches) to the upper position. Driving Variable High Path Dynamics Now, under the influence of a high Bistable Variable the far left position of Driving Variable is no longer stable, and it is driven to the right. At some critical point the Bistable Variable avalanches to the lower position. The more Driving Variable is driven to the right the more unstable the Bistable Variable becomes in the upper position.

14. Hysteresis as Bistable Behavior High Bistable Variable Low Driving Variable Description Two The System can exist in distinctly different states under the same conditions. System begins here; high Bistable, low Driving Driving Variable is driven to the right (by an external variable) Here Driving has small effects on Bistable. But system is building to critical state. Tipping Point: small change in Driving has large effect on Bistable. In here there are no stable states System now driven left

15. Hysteresis as Bistable Behavior Description Two The System can exist in distinctly different states under the same conditions. If one variable comes to dominate the system either goes run away positive feedback (chaos), or runaway negative feedback (closes down to point attractor) Increase external driving force and system bifurcates to different attractor Lower external driving force and system closes down System now driven left

16. Hysteresis as Bistable Behavior High Bistable Variable Low Driving Variable Description Two The System can exist in distinctly different states under the same conditions. System begins here; high Bistable, low Driving Driving is driven to the right (by an external variable) Here Driving has small effects on Bistable. But system is building to critical state. Tipping Point: small change in Driving has large effect on Bistable. In here there are no stable states Tipping Point: So system must avalanche down to lower path. System cannot return to upper path from here but must be driven left (because high One is not stable under high Two). System now driven left

17. Hysteresis Caveats: warnings and cautions Hysteresis loop describes the behavior of the system . . . State 1 It does not explain the cause-effect relationships behind the behavior. Bistable Variab le And it does not explain the source of the energy. State 2 These systems are driven by processes and energy sources outside the diagram. Driving Variable • Ultimately solar or tectonic energy. • The hysteresis Driving Variable is itself being driven. • Or, “everything is connected with everything else by positive and negative feedback.” • We can isolate the bistable system to discern the relationships among the variables, but what keeps the system “open” with enough “r” value comes from outside the system. Return to Systems

18. Hysteresis as Bistable Behavior Description Three The System can exist in distinctly different states under the same conditions. Social/Economic Example Paul Ormerod

19. Hysteresis Social/Economic Example How the proportion of criminals in a population varies with the level of social and economic deprivation Begin here: high deprivation, high crime For any given reduction in deprivation the impact on crime becomes stronger. But, not until the tipping point does the frequency of crime drop dramatically. Percentage of population who become criminals Tipping Point: Level of social and economic deprivation At the tipping point even a very small further reduction in the level of deprivation leads to dramatically lower crime rates. Once we are on the bottom line, additional falls in deprivation reduce crime by only small amounts.

20. Hysteresis Social/Economic Example How the proportion of criminals in a population varies with the level of social and economic deprivation The bistable nature is illustrated by today when the country is rather prosperous, but crime levels are very high, in contrast with the depression, when deprivation was very high, but crime was low. Liberal Explanation At the tipping point the level of deprivation is high enough that lots of people turn to crime just to survive Percentage of population who are criminals Tipping Point: Severity of Criminal Justice System Once on the bottom line crime remains low, or rises only very slowly even as the level of economic deprivation continues to rise.

21. Hysteresis Social/Economic Example How the severity of the criminal justice system affects the level of crime in the population. We assume that the effect of a more punitive criminal justice system is to reduce the proportion of criminals in the population. As the criminal justice system becomes more strict the proportion of criminals in the population falls. But the effect at first is rather minimal. Conservative Explanation Now the cost of crime is so low many people return to crime (out of greed?). At Tipping Point punishment is high enough it dissuades non-hard-core criminals from committing crimes. Percentage of population who are criminals Crime rates fall dramatically. As punishment laxes crime slowly increases (because crime has low cost) Severity of Criminal Justice

22. Hysteresis Social/Economic Example Simple Cause-Effect Relationship Bistable Relationship Return to Systems

23. Bistable Behavior in Bifurcation Diagrams A Geological Example Glacial/Interglacial Cycles

24. Bistable Behavior in Bifurcation Diagrams A Geological Example Glacial/Interglacial Cycles Cooling Part of Cycle High Weathering removes CO2 When continents are exposed, abundant weathering of exposed rock sucks down CO2 from atmosphere, slowly at first but with increasing effect with time. Earth Temperature • Lower CO2 concentrations in atmosphere lowers the greenhouse effect of CO2 leading to cooling. Low High Atmospheric CO2 Low • When it is cool enough for ice sheets to begin forming the increasing albedo causes temp-erature to drop even more. Albedo (ice reflection of sunlight) becomes a positive feedback system driving Earth deeper and deeper into the ice age.

25. Bistable Behavior in Bifurcation Diagrams A Geological Example Glacial/Interglacial Cycles Warming Part of Cycle High Weathering removes CO2 Expanded ice sheets lead to rise of CO2 in atmosphere. Rapid rise in temperature • Ice sheets decrease area of exposed rock, reducing weathering rates, decreasing loss of CO2 from atmosphere. Earth Temperature • Ongoing volcanic outgassing puts CO2 back into atmosphere. Low High Atmospheric CO2 Low At first these trends have minimal effect on Earth Temperature. But, when CO2 rises high enough its concentration in the atmosphere crosses a threshold leading to rapid warming trends leading to glacial melting. Return to Systems

26. Bistable Behavior in Bifurcation Diagrams A Geological Example Glacial/Interglacial Cycles Bard E. Abrupt climate changes over millennial time scales: climate shock. Physics Today 55, 32-37 (2002) (link) Return to Systems http://www.cerege.fr/tracorga/BARD/bard89.html

27. Hysteresis as Bistable Behavior Description Three The System can exist in distinctly different states under the same conditions. Lotka-Volterra Population Model Rising rabbit population causes fox population to rise. Rising fox population eats more rabbits causing rabbit population to fall. Falling rabbit population leads to fox starvation so their population falls. http://www.adeptscience.co.uk/products/mathsim/maple/powertools/des/unit26.html

28. Bistable Behavior in Bifurcation Diagrams Description Three Bifurcation Diagrams and Coupled Oscillating Variables Description Three

29. Bistable Behavior in Bifurcation Diagrams Under Symmetrical Conditions The system must either “choose” the upper path, increasing in X Which means the system cannot continue on in a straight line; that is not stable Threshold OR, the system must “choose” the lower path, decreasing in X lambda equivalent to “r” For lambda values less than lambdac there is only one value of X For lambda greater than lambdac there are two possible paths each of equal likelihood How does it choose in this situation?

30. Bistable Behavior in Bifurcation Diagrams Sensitive Dependence Which path the system follows is sensitive dependent and the outcome is unpredictable. Beyond lambda-c the system perceives differences at the microscopic level that would be insignificant at equilibrium and there is no way the system's evolution can be controlled, or directed, or predicted. Run Xnext

31. Bistable Behavior in Bifurcation Diagrams Under Asymmetrical Conditions And, this path is not possible, or is highly unlikely This path is not stable This is the only path the system can take For lambda values less than lambdac there is only one stable state of X For lambda greater than lambdac there are three possible paths not of equal likelihood The conditions of the system drive the system to a single solution; down a single path.

32. X shifts down This X value favors path (a) This X value favors path (b) Bistable Behavior in Bifurcation Diagrams When variable X is shifted up path “a” is preferred When variable X is shifted down path “b” is preferred External Variable Paths “a” and “b” on the bifurcation diagram are the upper and lower paths of the hysteresis diagram, as seen superimposed here. Old path no longer possible under new X conditions Hysteresis Curve

33. Bistable Behavior in Bifurcation Diagrams Lower Path Dynamics High X X and lambda are coupled in a positive/negative feedback system. What ever is driving lambda to the right . . . . . . The more lambda moves to the right the more X becomes unstable in the lower position. Low X At some critical (sensitive dependent) state X shifts (avalanches) to the upper position. Lambda High Path Dynamics Now, under the influence of high X the far right position of lambda is no longer stable, and it is driven to the left. At some critical point X avalanches to the lower position. The more lambda is driven to the left the more unstable X becomes in the upper position. Return to Systems

34. Bistable Behavior in Bifurcation Diagrams The system has to avalanche straight up or down on the hysteresis diagram since at these lambda values either the upper or lower path are possible (depending on X) but not the states in between.