The nonlinear nature of relationships is embodied in the dynamics of period doubling. In our little Snowflake circle, this "5th" attractor is the penultimate, anti-climatic crescendo in the journey to the next phase space. Staying coherent through it is essential before all the fun math ensues and a living system shifts from the exponential tension of complexity to the spontaneity of the strange attractor and true chaotic behavior. I'll walk the fractal terrain and discuss how we can better use the nature of period doubling on an intrapersonal, interpersonal, memetic and narrative level. My aim is to reframe how we understand identity and face uncertainty in both the present moment and our speculative futures so we can learn to recognize these dynamics in real-time and use them fully.
I originally asked the Snowflakes this back in 1998...In 2012, I have my answer...NO!
But COHERENCE is...lemme splain it...(again)
Clear As Mud...
When we place 'health' in a NDS context, the damn word just doesn't belong. When living systems (like us) demonstrate effortless or rhythmic growth and the ability to change [adapt] with increasing complexity as they develop over time, the onlookers call that health-y. The short story is that robust systems are resource-full. But the one constant in all this mess is that health is not an individual achievement. It's just an individuated process. The interaction we have with our surroundings shapes both our health potential and our capacity to recover it when it's been challenged. Using the dynamics of period doubling, we can move through this paradox of sustaining our subjective sense of constantcy while we explore the novel and uncertain aspects of daily life.
It's all relative...even in a dysfunctional family like the NDS community!
Period doubling is one of those terms that straddles the mathematical & metaphorical realm of the NDS lexicon. Check it out.
Some of the metaphorical ones include...
Edge of Chaos A term made popular by researchers at the Santa Fe Institute to indicate a particularly "pregnant" phase in the evolution of a dynamical, complex system where creative emergence of new structures is at a maximum. In the study of the behavior of cellular automata and similar electronic arrays, the edge of chaos seems particularly favorable for the emergence of innovative, more adaptive structures and modes of functioning. The edge of chaos is conceived as that zone between too much rigidity and too much laxity. There is controversy whether natural systems have a tendency to evolve into edge of chaos conditions. The edge of chaos can also be considered as roughly analogous to far-from-equilibrium conditions in that they both represent critical thresholds where self-organization and emergence are heightened. Organizational applications have to do with processes that encourage organizational innovation by facilitating edge of chaos like conditions.
Emergence The arising of new, unexpected structures, patterns, or processes in a self-organizing system. These emergents can be understood as existing on a higher level than the lower level components from which the emergents emerged. Emergents seem to have a life of their own with their own rules, laws, and possibilities unlike the lower level components. The term was first used by the nineteenth century philosopher G.H.Lewes and came into greater currency in the scientific and philosophical movement known as Emergent Evolutionism in the 1920's and 1930's. In an important respect the work connected with the Santa Fe Institute and similar facilities represents a more powerful way of investigating emergent phenomena. In organizations, emergent phenomena are happening ubiquitously yet their significance can be downplayed by control mechanisms grounded in the officially sanctioned corporate hierarchy. One of the keys for leaders from complex systems theory is how to facilitate emergent structures and take advantage of the ones that occur spontaneously.
Period-halving bifurcations leading to order, followed by period doubling bifurcations leading to chaos. (wiki)
Diagram of cusp catastrophe, showing curves (brown, red) of x satisfying dV/dx = 0 for parameters (a,b), drawn for parameter b continuously varied, for several values of parameter a. Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x givinga local maximum.
Both sides of the math-metaphor mirror have generated a swarm of buzz words that make deciphering these dynamics cumbersome...Everything from dissipative structures, to tipping points, to divergence, to the dreaded sepatrix and my personal favorite - subharmonic hysteresis have period doubling in their list of key ingredients. But eventually...after all the dust settles, we all seem to end up back at the ranch...staring at this thing!
Check out his Blue Sky Catastrophe (Fred & Martin - Can figure it this one out?)
I'd like to bridge these views by evoking the words of the great chaos philosopher
Lawrence Peter "Yogi" Berra who simply said...
"When you get to the fork in the road, take it."
Let's take one last look at this thing...there's some clues in this math fog.
"The line indicates that the attractor involved in the asymptotic dynamics is a fixed point, two lines indicate a periodic dynamics with oscillations among two constant values. The transition from the line to the double line is called period-doubling, and is one of the routes to the chaotic behavior. A successive number of period-doublings brings to the chaotic domain, which is represented by a cloud of disperse points. We can also see that in between the chaos can appear periodic windows indicating and ordered and predictable dynamics. Notice that the value of k initially used i.e. 1.9, falls in the chaotic region." (hmmmm...OK, go on....)
"The initial debate because of May's papers were if the apparent random fluctuations and the unpredictability in natural ecosystems may actually be due to deterministic chaos. Berryman and Millstein argued that in the bifurcation map of the logisitc equation (see Fig. 14), the population spends more time at extremely low densities, where there is a higher probability for deterministic extinction given for example because of external noise. In populations with small size the probability of extinction once the chaotic domain is reached is extremely high. They also said that living systems are antichaotic and that populations could enter in the chaotic domain because of the human action i.e. perturbations. This paper, with sentences like "deterministic chaos ... is more illusory than scientific ..." , actually provoked an intensive debate. Since then, lots of papers have shown that chaos do not involve a higher probability of extinction. The role of chaos as a stabilizing factor has been pointed out by Kaneko and Ikegami, who standed out a high dimensional, weak chaotic flow responsible to give stability in host-parasite population dynamics. Such chaotic flow, characterized by near-zero, positive Lyapunov exponents has been labeled homeochaos." (great name for a garage band too btw!)
OK, I think we've exhausted the contradictions surrounding the role of period-doubling. But to me, the reason why they never get resolved is that we try to place ourselves outside the system(s) we're observing. We study them as fossils frozen in time and not as the fluid, dynamic entities they (and we) are.
The same goes for health. We reduce it to numbers or catch phrases to remove the 1st person experience from it. Phooey!
Let's look at some different ways to perceive period-doubling that allow us to access that 'moment of truth' more. Then we can get past the linear, time-bound maps we've been given...
In Chaos & Complexity, author Michael Butz attacks the paradox from both fronts. First he clears up the fact that period-doubling goes in both directions! It can lead back to stability or it can lead to chaos. In his model, it's more a bridge between those extremes than a transitional state in and of itself. What I like about his approach is that it's truly NON-linear. He speaks of cycles and loops and an ebb and flow from order to chaos and back to order. That's much closer to our experience of health, self and life. They are are 'subject to change'.
Transition is the focus of Dai William's article in the journal Futures. Again, the nice thing about this approach is that it tracks multiple paths through the phases and views the period-doubling part as a choice point. So here we earn transformation by facing transition coherently... Simpler math!
"The crisis phase is a time for caution and mutual support. Individual defining moments can lead to the recovery phase - a time of rising morale, new insights and new initiatives. Greater awareness of transitions may be a key to personal and national survival in an era of unprecedented change."
For those of you who remember, George Muhs DC and I did a little DVD called "Nothing Personal" about the somatic aspects of dyadic communication back in 2003. Sigh...We described the shifting of attractors as a process where self and other essentially becoming one. When we reach period-doubling, choices are made to determine if the relationship evolves or dissolves...
"The critical attractor in dyadic communication is the fifth one, termed period doubling. It is depicted here as a large purple circle, representing the expansion of the blending of the red and blue circles. The exchange peaks here and the two parties determine how they'll characterize the conversation. If the blend was positive, the relationship deepens. If it was combative, it reverts back a few attractors. The term period doubling accurately describes what's happened. Both sides have twice the 'self'!" _____________________________________________________________________
It's Kramer vs. Shrek Baby...
I got in trouble with former SCTPLS president Dick Bird a lot. So much so that he intentionally tracked me down in 100+ degree heat in Madison, WI to scold me for not 'doing my math'. Sigh. Seems DB was a really big fan of entropy and conflict. I wanted to know more about enformy (conservation of organization) and cooperation. What he was really getting at was his very useful distiction of iteration and recursion. So props to him and here it is...
"To iterate means to repeat. However iteration should be distinguished from simple repetition, which may produce nothing. In the process of iteration a series of actions is carried out, the result of each action in the series becoming the object of the same action the next time it is performed… Recursion is related to iteration by a simple twist of definition. Instead of the result of an action being the object of the repeated action on the next occasion it is performed, as in iteration, in recursion the object of the action becomes itself; in other words the action becomes a self-referential one….Iteration takes place in a series of steps in time, while recursion takes place outside time or rather embraces all times in its definition. Iteration works forwards, by taking a value and then building on it to produce the next in a series; recursion, on the other hand, takes a value and defines it in terms of itself.. Recursion and iteration can be seen to be inverse processes rather like multiplication and division, or addition and subtraction. What is inverted is time: iteration takes place in time as it moves forward, while recursion involves an apparent backward step in time, implied by the element of self-reference in its definition." http://www.aber.ac.uk/tfts/journal/current/bird.html
But what it really boils down to for me is how you enter into the experience of birfurcation. Do you shift or do you sh*t? :)
In my original research paper (Filippi, 1999) I used a quote from Margaret J. Wheatley's very accessible book Leadership and the New Science "What we observe, in ourselves and as well in all living entities, are boundaries that preserve us from and connect us to the infinite complexity of the outside world. Autopoesis, then, points to a different universe. Not the fragile, fragmented world we attempt to hold together, but a universe rich in processes that support growth and coherence, individuality and community.”
I later quoted our own Terry Marks-Tarlow on the same vibe...“If psychological health resides at the edge of chaos, then our infinitely deep fractal core may be as much a source of renewal, inspiration, and creativity
as a source of confusion and pathology.”
So if we make the turn here, if you're interested in structure you frame period-doubling by iteration's rules and look for the exceptions to self that lead to 'what's next'. If you're interested in function you frame it by recursion's rules and look for ways to expand the self in the moment. Levels or layers?
"Levels, Jerry... levels!"
"Layers! Onions have layers. Ogres have layers.
You get it? We both have layers."
I'll tie this thought up with another graphic from "Chaos & Complexity"
We're now off the flat page of math. We've left the fuzzy clouds of metaphors. We've seen how we can view period-doubling and health as an inside-out, 1st person experience. So each time we face the choice period-doubling presents us we can use the uncertainty of the moment to let more novelty OUT. Instead of fighting our environment, we can feed off of it.
Using & Reusing Period Doubling Soma Style...
On an intrapersonallevel we can use any number of topological maps of the body to tap into the layers of somatic recall we all possess. For period-doubling, we focus on expanding awareness of the hands and how they can alter our experience of our energy, balance and focus. By adding multiple connotations, we create an experiential anatomy that gives us more ways to naturally adapt to and face uncertainty.
On an interpersonallevel we can use the nonverbal nature of somatic exchange to tap into the layers of neurobiological circuits we use to regulate reactivity. For period-doubling, we focus on expanding awareness of our cognitive and visceral dissonance. In that way, we connect our instinctive enteric nervous system to our central nervous system and create a social congrency that Stephen Porges calls "stage 3 dyadic communication' which fosters deeper human bonding.
If you'd like to experiment further with these principles, I wrote an e-book in 2004 called The YES Factors which applied what Porges wrote about with the somatic functions as detailed by Thomas Hanna. It dovetails with period-doubling nicely as see in the chart below. The folding over of the practices feeds both the recursive and iterative aspects of managing birfurcations coherently. Fractals and solitons unite!
On a memeticlevel we can use Clare Graves' model of vMemes that was refined by Spiral Dynamics to access the layers of social conditioning we absorb as we grow and develop. For period-doubling, the key aspect here is that we can combine the recursive and iterative aspects of period-doubling by applying the principle Ken Wilber called 'transcend and include'...
"The warm-colored spirals (beige, red, orange, and yellow) focus on personal experience, defining yourself and your individuality. The cool-colored spirals (purple, blue, green, and turquoise) focus on community and group experience, finding your place in and fitting into the whole."(link)
On an narrativelevel we can use chronologically-oriented timelines to visually unpack the layers of undifferenciated kyros (moments of decision/indecision) we all experience as a felt-sense. For period-doubling, we focus on expanding awareness of various time frames, especially those concerned with present to future (goals) and future to future (speculation/wishes) as we crossover from thought to action. This helps us recognize where our intentions are rooted. Moving our attention off the 'Immediate Self' to the "Extended Self" and then into the specific spatial locations of these time frames is a powerful somatic tool for us to entrain. In this way we can move beyond the body, outside the mind and into an open identity. But only the soma is suited to carry us there...
"The self is a reflection of the soul in matter." Fred Alan Wolf
"The self is a circle whose center is everywhere
and whose circumference is nowhere." Carl Jung
"The self is rediscovered whenever the mind and body agree." Mark Filippi
Every living system experiences stability, instability and transition. This is called the SIT concept. Stability is our sense of immortality or the higher self, the timeless core of our being. Instability is our experience of our mortality through the ebb and flow of relationships, which is where your extended self emerges from. The transition occurs in the medium of reality, through the survival-based actions of the lower self.
#1 So are you...experienced? And have you been experienced?
The dynamics of period-doubling allow us to enter an awareness that math and metaphor fail to capture completely. It lets us view health as a byproduct of adaptation that robust living systems make when faced with uncertainty. Being able to navigate these choice points or moments of truth determines how successfully we transition from the status quo of our established closed identity to the challenges of growth and development that leads to the spread of social coherence.
#2 If you're gonna be NON-linear, stop using linear stuff!
Kerplunk! Health, self and life do not subscribe to the constraints we place on them. We live in a wiggly world that requires a more flexible, portable, and most of all, vitalistic set of resources so we can handle the turbulence that leads to transformation, transcendence and tensegrity! Can you dig it?
"The HeartWave is a combination of the body's energy output during brief exercise and recovery, the large wave, and a representation of the heart contracting and relaxing (systole and diastole), the small wave superimposed on the larger wave. The third set of waves, the smallest, represents the biochemical oscillations of the heart function. This is not a conventional graph, with a single time dimension along the bottom. The bottom line is the time dimension for the large wave of exercise/recovery; the time dimension for the systole/diastole waves is the exercise/recovery wave; and the systole/diastole wave is the time dimension for the biochemical oscillations."
#3 Remember Plexus?
We had a chance as a group about 10 or 12 years ago to align the NDS model with health care...and we completely dropped the chalupa. What ended up happening was another think tank of theorists filtered their ideas into the mainstream medical community and promptly got marginalized. So ideas like Irving Dardik's SuperWave (above) were sent packing into the fringe, along with others (hi!). Stop waiting for the likes of Health 2.0, Obamacare or any group to usher in these underutilized, highly effective ways to improve health potential.
#4 One for the road...
Recently my noble profession won a small battle with the Department of Education to thwart the efforts of an entity called the Counsil on Chiropractic Education, Commission on Accreditation efforts to remove the vitalistic elements from the curriculum. Hot off the presses, the CCE seems to have found a loophole to skirt the sanctions imposed on them. It appears the math (ie. lying with statistics) has masked the mindset of the field again. Why do I mention it? It's just another example of a moment of truth being nearby. The lack of coherence of the keepers of the flame here may be the death knell for a model that NDS principles could of explained to anyone easily.
The explanations for period-doubling's dynamics are very similar to the ways we evaluate health. The top picture is from the mathy-math of Larry Liebovitch's ppt "Introduction To Fractals" which floats something called the Shadow Theorum (Google it) to maintain the idea that primacy is with approximations and equations despite its flaws. The bottom one illustrates the integral world's perception that embraces a metaphor that embodies all the levels, lines, labels and layers into one living whole. That can strain the imagination, even for sapes like me. I ask both camps to consider using the soma to bridge these abstract extremes so we can harvest the fruits of 40 years+ of theorizing. Let the fractals and solitons play!