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Chaos theory has been proved to be of great significance in a series of critical applications although, until now, its applications in analyzing soil respiration have not been addressed. This study aims to introduce a chaotic component in the control system of soil respiration and explain control complexity of this nonlinear chaotic system. This also presents a theoretical framework for better understanding chaotic components of soil respiration in arid land. A concept model of processes and mechanisms associated with subterranean CO_{2} evolution are developed, and dynamics of the chaotic system is characterized as an extended Riccati equation. Controls of soil respiration and kinetics of the chaotic system are interpreted and as a first attempt, control complexity of this nonlinear chaotic system is tackled by introducing a period-regulator in partitioning components of soil respiration.

Chaos is a kind of external, complex, and seemingly irregular motion in the deterministic system due to randomness [

Because the chaos system can produce “unpredictable” pseudo-random orbits, many research studies focus on the related algorithms and performance analysis of constructing pseudo-random number generators utilizing chaos systems. For continuous chaotic systems, many chaotic pseudo-random sequences have been proved to have excellent statistical properties. However, until now, applications of chaos theory in analyzing soil respiration have not been addressed. It is necessary to introduce a chaotic component in the control system of soil respiration and explain control complexity of this nonlinear chaotic system. In previous studies, we have found that soil respiration (_{s}) estimate in arid regions should not have neglected the contribution of abiotic exchange [_{2} sink [_{io}) is therefore necessary to be taken into account for a more reliable estimate of soil respiration in arid regions [_{io} and show that it is a chaotic component of soil respiration in arid regions.

Objectives of this study are (1) to show that _{io} is a chaotic component of soil respiration in arid land and present a theoretical framework for a better understanding of this chaotic component, (2) to interpret the chaotic system on controls of soil respiration and kinetics of the chaotic system, and (3) to reduce the control complexity of this nonlinear chaotic system by introducing a period regulator.

We hypothesize that the underground CO_{2} assignment in arid and semiarid regions has been regulated by a hidden loop in groundwater cycle. In brief, groundwater discharge and recharge have regulated the components of soil respiration. Based on this hypothesis, subsurface CO_{2} transportation, dissolution, sequestration, and other reassignment processes in the soil-groundwater system are largely driven by precipitation, evaporation, irrigation, dew deposition, etc. These are hydrologic processes associated with the chaotic component _{io}of soil respiration. Such processes regulate the storage and turnover rates of inorganic carbon and its dissolvable part in the profile of soils [_{2} in soil can react with dew and then dissolve carbonate or even migrate into saline aquifer [

Influenced by the hidden loop, soil respiration in arid regions is no longer a definite system. It becomes a nonlinear chaotic system. In order to describe the nonlinear chaotic system, the conceptual framework of known and unknown processes associated with the hidden loop in groundwater cycle, along with the possible mechanisms, is shown in Figure

Story of the hidden loop in the nonlinear chaotic system, including known and unknown parts.

The hypothesized hidden loop can explain particularity of CO_{2} assignment in arid and semiarid regions. Differential, difference, and dynamic equations are used for modeling many problems arising in engineering and natural sciences [_{2} is hypothetically dissolved in saline aquifers, we characterized the dynamics of CO_{2} concentration in the groundwater-soil system in [

However, there are still considerable uncertainties and difficulties in fully understanding the underlining mechanisms and critical factors driving such a hidden loop. One major challenge is how to characterize the structure of the soil-groundwater system [_{2} sequestration in different layers should be different. The whole story is shown in Figure

Hypothetical system kinetics of the hidden loop: (1) three carbon pools (the atmosphere, soil and groundwater) are connected through carbon cycles and water cycles, along with the underlining processes associated with CO_{2} sinks (green solid circles) and sources (blue solid circles); (2) the inorganic CO_{2} change beyond the red rectangle (if excluding influences of groundwater) are driven by evapotranspiration and vapor condenses, while the inorganic CO_{2} assignment and ventilation within the red rectangle are largely driven by groundwater recharging/discharging.

In previous publications, it was demonstrated that the variations of _{io} originate from the physical forcing of abiotic factors such as soil salinity (EC), alkalinity (pH), temperature (_{s}), and water content (WC_{s}) and their linear relationships with its daily mean intensity appear to be valid within a seasonal cycle as a whole. However, in diurnal cycles, taking into account the complicated and undetermined processes associated with the chaotic component _{io}, the soil respiration system in arid land is a nonlinear chaotic system. Variability in the data of _{io} presents further evidence for _{io} being chaotic. Before the chaos theory was proposed, scientists had thought that there are only two kinds of phenomena—the phenomena which act strictly according to a rule and the phenomena which happen stochastically [_{io} (Figure _{io} are seen to interact (Figures _{io} looks stochastic (Figure _{io} in diurnal cycles is an exponent-sine coupled normalization transformation of time sequence (TSN), as follows:

The period character of CVPC (a) almost coincides with the period character in the hourly scale variations of _{io} (b). None of _{s} (c), WC_{s} (d), and the optimal linear combination of _{s} and WC_{s} (e) is better than CVPC (f) to describe the temporal pattern of _{io} in diurnal cycles.

Since soil respiration in arid land is a nonlinear chaotic system, the resulted control complexity is naturally reconciled [_{10}) of _{s}. Analyses on data collected from previous studies revealed diel turbulence in _{10} values even if excluding the negative _{s} data. On the basis of utilizing the basic and reanalyzed data collected from [_{10} values is far from certain. All the _{10} values used in the analysis were calculated utilizing the simple model of _{s} (the derivative of the exponential chemical reaction-temperature equation originally developed by Van’t Hoff) [_{s} were not included in calculations of _{10}. Controls of _{10} at each site were, respectively, analyzed in linear regressions for a between-ecosystem comparison. Results from these analyses were further compared with the analyses of the variation of _{10} with _{10} values from both sites, the effects of WC_{s} on the _{10} of _{s} to _{s} and the _{10} of _{s} to _{a} were analyzed in quadratic regressions. In order to further test the role of WC_{s} in determining _{10}, four coupling models were employed to analyze coupling effects of _{s} on _{10}. The front two models were established under the hypothesis that the influences of WC_{s} and _{10} were mutually independent. The first model hypothesized that the influences of WC_{s} and _{s} and _{s} and _{10} were not mutually independent. The third model hypothesized that _{10} was dominantly determined by WC_{s} and _{10} to WC_{s}; the fourth model hypothesized that _{10} was dominantly determined by _{s} linearly interacted on the responses of _{10} to

We further examined the variability of _{10} values, as seen in Figures

Diel turbulence in temperature sensitivities (_{10}) with soil temperature (_{s}) and water content (WC_{s}).

Diel turbulence in temperature sensitivities (_{10}) with air temperature (_{s}) and water content (WC_{s}).

Taking into account negative _{s} data in arid regions is strongly necessary to reduce uncertainties in the current global/regional carbon balance and in the predictions of future feedbacks in the coupled carbon-climate system ([_{2} footprints ([_{io} (being averaged among diverse soil sites and meanwhile averaged from different days) by the linear combination of TS, WCs, and CVPC. Let _{io} can be easily extended to daily or larger scales.

Utilizing the data in Figure

Treating the control complexity by the proposed model on subsequent days after rainfalls (a1–a4, c1–c4), which were modified in simulations by equation (

Due to potential overlap in environmental, temporal, and spatial components of ecological data, partitioning the variations among pure environmental controls, pure spatial controls, pure temporal controls, pure spatial component of environmental controls, pure temporal component of environmental controls, pure combined spatial and temporal component controls, combined temporal and spatial components of environmental controls, and unexplained component should be included in multivariate analysis of the chaotic system. The whole story of control complexity of this nonlinear chaotic system is therefore worthy of further investigation.

In reference [_{io} of soil respiration and characterize the dynamic of CO_{2} concentration in the soil-groundwater system as an input-output balance equation, as follows:_{2} concentration in a considered gas room _{2} concentration in the atmosphere. For the _{n} is the average ratio between the input and output of CO_{2}.

Suppose that the input of CO_{2} into the soil-groundwater system was finally dissolved in the groundwater of volume _{io} are driven by groundwater discharge (outflow) and recharge (inflow), with volume

Finally, considering the restricting effect of current DIC, which is characterized as

The next research priority is to analyze the characteristics of bifurcation and chaos in the inherent spatial and temporal variations of _{io} by using Feigenbaum graphs [_{2} is the third determining process of _{io} besides the input and output of CO_{2}, which involves organic components of soil respiration. This process, along with the input and output of CO_{2}, determine the increase rate _{2} concentration and also determine the density of _{io}.

For a better understanding of how soil CO_{2} fluxes change with space and time, it is necessary to introduce _{io} as a nonlinear chaotic component of soil respiration in arid land. Ecology is a study not how things but how things change with space and time, and hence, it is also necessary to interpret the control complexity of this chaotic component. In the assessment of the importance of organic and inorganic factors influencing _{io}, inherent spatial and temporal variations in ecological data should be taken into account whenever possible. A next research priority is to analyze the characteristics of bifurcation and the chaos difference between the subterranean and surficial CO_{2} concentration and further understand the whole story of the control complexity of _{io}.

All the data utilized to support the theory and models of the present study are available from the corresponding authors upon request.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This research was funded by the National Natural Science Foundation of China (41571299) and the High-Level Base-Building Project for Industrial Technology Innovation (1021GN204005-A06).

_{2}efflux

_{2}flux and evapotranspiration in a Chihuahuan desert grassland

_{2}absorption

_{2}sink

_{2}budget