Abstract In this paper, brief annotation of a constructing procedure and the applications of connection equation are given. An analytical method of the connection equation as an approximate equivalent analytical solution of nonlinear dynamics equations is discussed. Research direction of Jiansics or Dimennics about spectrum(spectrum array) formed by series of laws and their connections in evolution processes of natural phenomena are explored, an equation of the transformation of the phenomena between the states in evolutionary process is presented, that it follows a natural optimum principle or a natural conciseness principle in general, which having an equation form of the smooth curve with the optimal path or the most concise path, simple constructing forms of forward partial-symmetrical equation(or weak-symmetrical equation) and backward partial-symmetrical equation are given, an approximate expression of extended hyperbolic tangent series for data curve or partial continuous functions is constructed, then natural evolution-balance rule is presented, and tendency equation of biological growth process is given too. According to approximate form of the nonlinear dynamics equation of the connection equation, two new electronic circuit elements(nonlinstor and geomsentor) with deepening charge-controlled capacitor properties based on form of the nonlinear differential equation is predicted, and the nonlinear differential equation for a RLCNG series circuit is also analyzed, extended direction of the general distribution function and the general distribution density function, tendency equation and its conditional solutions of the statistical distributions of the particles are given, properties of the frequency interval of Planck’s quantum equation are explored, nonlinear differential equation expressions with solution of approximate linear frequency conversion and their characteristics of curve shapes of frequency conversion wave equations at negative frequency are discussed, a concise model of database theoretical framework (this framework to be made up of foundation database, tendency equation, and analytic database) is explored, an equation of relationship between the total annual energy consumption with the annual GDP in the United States, and an equation of relationship between the annual population with the annual GDP in the United Kingdom are established, and limit values of the total annual energy consumption in the United States and the annual population in the United Kingdom are calculated and predicted. Subsequently, tendency fitting equations of curves are explored, which included the creep process curve of the rock and the single-crystal superalloy, the Volt-Ampere characteristic curve of the discrete semiconductor device, the resistance (or resistivity )-absolute temperature curve of the superconducting material, the direct current I-U characteristic curve of the bicrystal Josephson junction, and the current step amplitude of Shapiro steps, the friction-speed characteristic curve (that included the stages of Coulomb friction, Stribeck friction, viscous friction, friction hysteresis, and anomalous friction hysteresis effect) in mechanical system or servo system, an expanded equation form of the Newton's law of cooling, the shear stress-shear rate in the flow curve (that included the sections of first Newtonian fluid-flow, pseudoplastic flow, second Newtonian fluid-flow, dilatant flow, turbulent flow, and anomalous shearing effect) of the polymer melts, etc. On phenomenological and tendency levels, research directions of similar property of creep curves of materials with the flow curves of polymer melt, and similarity in the fractal measure of the evolutions of fracture cracks growth structure and fluid turbulent eddies nested structure are pointed out. At the end, limitations of the connection equation in data fitting and long-range forecasting are discussed.
Keywords connection equation, nonlinear dynamics equation, spectrum array of laws, Jiansics or Dimennics, states transforming equation, natural conciseness principle, natural evolution-balance rule
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