Integer-dimensional fractals of nonlinear dynamics, control mechanisms, and physical implications

Abstract

Fractal dimensionality is accepted as a measure of complexity for systems that cannot be described by integer dimensions. However, fractal control mechanisms, physical implications, and relations to nonlinear dynamics have not yet been fully clarified. Herein we explore these issues in a spacetime using a nonlinear integrated model derived by applying Newton’s second law into self-regulating systems. We discover that (i) a stochastic stable fixed point exhibits self-similarity and long-term memory, while a deterministic stable fixed point usually only exhibits self-similarity, if our observation scale is large enough; (ii) stochastic/deterministic period cycles and chaos only exhibit long-term memory, but also self-similarity for even restorative delays; (iii) fractal level of a stable fixed point is controlled primarily by the wave indicators that reflect the relative strength of extrinsic to intrinsic forces: a larger absolute slope (smaller amplitude) indicator leads to higher positive dependence (self-similarity), and a relatively large amplitude indicator or an even restorative delay could make the dependence oscillate; and (iv) fractal levels of period cycles and chaos rely on the intrinsic resistance, restoration, and regulative delays. Our findings suggest that fractals of self-regulating systems can be measured by integer dimensions.

The control mechanisms of heart rate dynamics in a new heart rate nonlinear time series model

“…(ii) HR fluctuation depended on two factors: cardiac disturbances and responses, but not a single factor such as cardiac autonomic modulation. It has been demonstrated that wave slope and amplitude depend on the relative strength of external disturbances to internal responses respectively 36 . Based on the wave indicators, HR slope indicator equaled the ratio of the mean of myocardial disturbances to the myocardial electrical resistance coefficient; HR amplitude indicator equaled the ratio of the standard deviation of myocardial disturbances to the relative strength of PNS against SNS activity (PNS-SNS activity).…”

“…denotes the integer part of T∕m) 36 . Consider the observation scales ranging from 1 to 10 seconds and from 1 to 10 minutes and real f1y02, f1o04, chf01, s20641, and 0030.vf1 (which were regarded as the representative heart rate time series of the YOUNG, ELDER, CHF, ST, and VT/VF groups, respectively) on the first scale m = 1.…”

“…Heart rate is controlled by a stochastic self-restoring mechanism. In recent years, there has been a growing interest in the nonlinear autoregressive integrated (NLARI) process derived by applying Newton's second law to stochastic self-restoring systems [34][35][36][37][38] . The NLARI process can exhibit the main HR features as mentioned above.…”

“…This implies that η 2 = σ∕β measures the wave amplitude. More importantly, the two wave indicators are also the fractal indicators that determine whether fractal behavior occurs and control the fractal level 36 . Usually, a relatively large absolute slope indicator reflects the high level of dependence, while a relatively small amplitude indicator reflects the high level of self-similarity.…”

Fractal Analysis of Strain-Induced Microstructures in Metals

“…The most common one is the fractal dimension. Fractal dimensionality is considered a measure of complexity for systems that do not present integer dimensions [25]. However, there are different fractals that have the same fractal dimension.…”