surface. More generally, changes in this valve formation system exhibit ergodic behavior, and the stability of such behavior in a dynamical system is measurable as Lyapunov exponents.
In a dynamical system, different states occupy different spaces that exist at different times. Changes from state to state can be delineated using a system of differential equations derived from the original relation among the variables of interest. A Lyapunov function of the general form x i = fi (x) has first derivatives that form the elements of the Jacobian matrix (i.e., Jacobian)
The relation between entropy and symmetry is of the general form of xi = f(xi), and from the Boltzmann entropy equation, xi = Si and f(xi) = -ln wi A Lyapunov function, Si = - ln w for each ith state of n-symmetry states at S = 0, is a system of homogeneous differential equations representing the change in symmetry states over time as a dynamical morphogenetic system. In matrix form, S = AS is a time evolution equation where A is the matrix of constant coefficients from
To determine the behavior characteristics of symmetry states, solution to the Lyapunov function of the form ATS + SA = -Q is evaluated, where Q is the resultant matrix. If the left side of the equation is determined to be positive definite when the right side of the equation is negative definite, then the symmetry states system is stable; otherwise, the system is unstable [2.23].
Specific characteristics of instability are evaluated via A. At equilibrium, (A - 9I) = 0 and for det(A - 9I) = 0, 9 are eigenvalues of A, and tr
Lyapunov exponents [2.88, 2.101] are meaningfully characterized by the real parts of the complex valued eigenvalues of A [2.29] and are
Lyapunov exponents comprise a multidimensional spectrum indicative of the rate of (exponential) divergence of multiple symmetry states with respect to direction of changes from one symmetry state to the next. Lyapunov exponents give bounds on information production of a dynamical system. They measure local behavior of the collective properties of a dynamical system yielding a global characterization of the system [2.8, 2.133]. A Lyapunov function close to equilibrium ensures that entropy tends to move toward a stable maximum in a dynamical system [2.29].
Lyapunov exponents are invariant to coordinate transformations and are ordered from largest to smallest [2.29]. They are a measure of the behavior of vectors in a tangent space. As such, these exponents are a measure of stability or instability at each symmetry state. For Lyapunov exponents that are positive, the dynamical system is chaotically unstable [2.29, 2.118], while negative Lyapunov exponents are indicators of a stable dynamical system [2.29]. A Lyapunov exponent with a value of zero may indicate a dissipative, regular [2.35], weakly chaotic [2.73] or intermittent system [2.74], depending on the particulars defining the initial system.
By quantifying the relation between symmetry states and stability via Lyapunov exponents, instability is characterizable as a specific behavior, either deterministically chaotic or not. However, the picture may be more complicated. Slight changes in initial conditions can precipitate chaotic behavior resulting in instability [2.29] even though the initial conditions may be random. Chaotic systems may appear to be random when considering fewer than all possible states in the dynamical system [2.97]. High-dimensional systems may appear to be ordered despite non-convergent random behavior [2.35]. Different kinds of randomness as “noise” may appear to be chaotic in a given dynamical system [2.118]. Care must be taken to discern the behavior of a dynamical system in terms of randomness or chaoticity.
2.2.8 Randomness and Instability
Lyapunov exponents enable the assessment of randomness with respect to instability in a dynamical system [2.74, 2.118, 2.163]. While local unstable behavior on a state-by-state basis may be chaotic, the macro-state instability of valve formation may be inherently random at a global scale. Alternatively, there may be intermittent chaotic or random unstable states [2.74] during the morphogenetic process. Because of this, a test for randomness [2.34] is necessary.
Thus far, our dynamical system model has been analyzed at equilibrium. However, to detect possible random instability, we need to consider the possible non-equilibrium consequences of such behavior in this system. Non-equilibrium may be a long-term phenomenon in contrast to short-term steady-states present in a dynamical system, for example, as time oscillations create Turing effects of diffusion driven instability [2.6]. Toy models have been used to link time intervals over multiple levels enabling the application of equilibrium dynamics to non-equilibrium behavior [2.96]. They have great potential in modeling complicated biological processes [2.2]. Non-equilibrium dynamics can be assessed in terms of constrained information loss or an increase in entropy [2.65].
As entropy increases over longer and longer sequences for a given dynamical system, entropy blocks or sections that define the structure of these sequences do not characterize the entire system. What remains is randomness that is not taken into consideration and given as the source entropy rate
Maximizing the rate of variation in the entropy values during valve formation involves converting Boltzmann entropies of symmetry states to Kolmogorov-Sinai (KS) entropies based on probability [2.97]. KS entropies are time-averaged Shannon entropies over joint probability space [2.147] and are used to construct a PDF as a maximum entropy distribution [2.65, 2.137]. Randomness distributed over a probability distribution involves maximum entropy with regard to a PDF.
Evaluating entropy S using the PDF of associated probabilities involves first derivatives of S as entropy rates [2.26, 2.74] corresponding to bandwidth in the histogram used to determine the PDF [2.137]. For the expected value of the probabilities associated to
