| Summary: | We study the general property of the evolution of a class of scalar fields with tracking and thawing behaviors. For the tracking solutions, we show explicitly with three different potentials that, independent of initial conditions, there exists a general relation between the equation of state <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>w</mi><mi>ϕ</mi></msub></semantics></math></inline-formula> and the fractional energy density <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mo>Ω</mo><mi>ϕ</mi></msub></semantics></math></inline-formula>, so that the scalar field follows the same <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectory during the evolution. The analytical approximations of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> trajectories are derived even though the analytical expression depends upon the particular form of the potential. For thawing solutions, a universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation exists and the relation is independent of both the particular form of the potential and the initial condition of the scalar field. Based on the derived <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation for the thawing models, we derive a tighter upper limit on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>w</mi><mi>ϕ</mi><mo>′</mo></msubsup><mo>=</mo><mi>d</mi><msub><mi>w</mi><mi>ϕ</mi></msub><mo>/</mo><mi>d</mi><mo form="prefix">ln</mo><mi>a</mi></mrow></semantics></math></inline-formula>. The observational data is also used to constrain the thawing potential with the help of the universal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>w</mi><mi>ϕ</mi></msub><mo>−</mo><msub><mo>Ω</mo><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula> relation.
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