15.5 Catastrophes in Complex Systems
603
therefore survivability of dominant pathways. The threshold for this kind of change
in ordering is sometimes called ‘self-organized criticality’. 9
Biological systems have a large number of structured subsystems, and therefore
are more ordered than that of the thermodynamic equilibrium state of the system’s
entities. But in addition, biological systems have evolved dynamical pathways that
are far more likely than those that might be physically realizable but are largely
unexpressed. Catalytic proteins are examples of evolved agents. People are another.
Biological systems have self-organized complexity. The theory of ‘steadily directed
systems’ (see Sect. 9.25) describes the evolution of self-organized complexity using
the language of statistical mechanics.
The ‘rules’ for how DNA produces proteins is an example of an emergent
relationship among agents in a complex system. Rules that describe the network
of effective biochemical reactions in cells is an emergent set of relationships. The
behavior of a brain in reaction to a stimulus is yet another example.
As the complexity of a system evolves, the hierarchical ‘levels of complexity’
may become distinguishable. For example, for humans, atoms are the fundamental
particles in the ‘zeroth’ layer, macromolecules are the agents at the first level of
complexity, organelles in cells are within a second level, cells themselves in a third,
organs in a fourth, individuals in a fifth, and societies in a sixth. Each level has its
own emergent ‘rules’ for its dynamics. Thus, complexity has a measurable degree
and may have observable levels.
15.5 Catastrophes in Complex Systems
If you pick up a starving dog and make him prosperous, he will not bite you. This is the
principal difference between a dog and a man.
— Mark Twain
Open systems with non-linear dynamics can suddenly change state. We have
seen that chaotic dynamical systems show such behavior. Population dynamics
(Sect. 13.2) was our first example. A system as simple as an anharmonic oscillator
with a driving force that slowly increases in frequency can suddenly change in
amplitude near resonance. This behavior occurs in the solutions to d 2 y/dt 2 +
ady/dt +by +cy 3 = A sin ((t), which is Newton’s law for the damped motion of a
mass held by a non-linear spring, and forced to oscillate. Figure 15.4 shows how the
9 The creation of reproductive systems is not confined to molecular systems. In June, 1959, Lionel
Penrose and son Roger showed how to build wooden entities with hooks and springs, which, when
placed in a box and shaken, could link together and then, after reaching a certain configuration,
release subassemblies. Also, the evolution of new entities in a complex system can be demonstrated
within a seemingly simple solitary game devised by mathematician John Horton Conway and
published in Martin Gardner’s column in the October 1970 edition of Scientific American. The
computer program called ‘The Game of Life’ is based on Conway’s scheme.
603
therefore survivability of dominant pathways. The threshold for this kind of change
in ordering is sometimes called ‘self-organized criticality’. 9
Biological systems have a large number of structured subsystems, and therefore
are more ordered than that of the thermodynamic equilibrium state of the system’s
entities. But in addition, biological systems have evolved dynamical pathways that
are far more likely than those that might be physically realizable but are largely
unexpressed. Catalytic proteins are examples of evolved agents. People are another.
Biological systems have self-organized complexity. The theory of ‘steadily directed
systems’ (see Sect. 9.25) describes the evolution of self-organized complexity using
the language of statistical mechanics.
The ‘rules’ for how DNA produces proteins is an example of an emergent
relationship among agents in a complex system. Rules that describe the network
of effective biochemical reactions in cells is an emergent set of relationships. The
behavior of a brain in reaction to a stimulus is yet another example.
As the complexity of a system evolves, the hierarchical ‘levels of complexity’
may become distinguishable. For example, for humans, atoms are the fundamental
particles in the ‘zeroth’ layer, macromolecules are the agents at the first level of
complexity, organelles in cells are within a second level, cells themselves in a third,
organs in a fourth, individuals in a fifth, and societies in a sixth. Each level has its
own emergent ‘rules’ for its dynamics. Thus, complexity has a measurable degree
and may have observable levels.
15.5 Catastrophes in Complex Systems
If you pick up a starving dog and make him prosperous, he will not bite you. This is the
principal difference between a dog and a man.
— Mark Twain
Open systems with non-linear dynamics can suddenly change state. We have
seen that chaotic dynamical systems show such behavior. Population dynamics
(Sect. 13.2) was our first example. A system as simple as an anharmonic oscillator
with a driving force that slowly increases in frequency can suddenly change in
amplitude near resonance. This behavior occurs in the solutions to d 2 y/dt 2 +
ady/dt +by +cy 3 = A sin ((t), which is Newton’s law for the damped motion of a
mass held by a non-linear spring, and forced to oscillate. Figure 15.4 shows how the
9 The creation of reproductive systems is not confined to molecular systems. In June, 1959, Lionel
Penrose and son Roger showed how to build wooden entities with hooks and springs, which, when
placed in a box and shaken, could link together and then, after reaching a certain configuration,
release subassemblies. Also, the evolution of new entities in a complex system can be demonstrated
within a seemingly simple solitary game devised by mathematician John Horton Conway and
published in Martin Gardner’s column in the October 1970 edition of Scientific American. The
computer program called ‘The Game of Life’ is based on Conway’s scheme.