<p>This paper investigates the planar <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> dual Minkowski periodic problem, which is reduced to a second-order differential equation involving a periodic data function and indices <i>p</i>,&#xa0;<i>q</i>. We establish the existence of a positive periodic solution of the equation for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\( 0&lt; p &lt; q \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>p</mi> <mo>&lt;</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and prove the nonexistence of periodic solutions when <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( p = q \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, numerical bifurcation analysis is employed to validate the main theorem and further investigate the dynamics of periodic solutions as the average data varies. The analysis reveals that the equation exhibits a saddle-node bifurcation of periodic solutions for certain parameter values. This dynamic behavior illustrates the transitions among the nonexistence, existence and multiplicity of periodic solutions as the average data gradually increases, corresponding respectively to the nonexistence, uniqueness and non-uniqueness of convex bodies.</p>