On the second variation of the biharmonic Clifford torus in $$\mathbb S^4$$

The flat torus $${{\mathbb T}}=\mathbb S^1\left( \frac{1}{2} \right) \times \mathbb $$ admits a proper biharmonic isometric immersion into the unit 4-dimensional sphere $$\mathbb S^4$$ given by $$\Phi =i \circ \varphi , where $$\varphi :{{\mathbb T}}\rightarrow S^3(\frac{1}{\sqrt{2}})$$ is minimal Clifford and $$i:\mathbb S^3(\frac{1}{\sqrt{2}}) \rightarrow small hypersphere. first goal of this paper to compute index nullity . After, we shall study in detail kernel generalised Jacobi operator $$I_2^\Phi We prove that it contains direction which natural variation with vanishing first, second third derivatives, such fourth derivative negative. In part paper, analyse specific contribution context, more general composition $${\tilde{\Phi }}=i {\tilde{\varphi }}$$ $${\tilde{\varphi }}: M^m S^{n-1}(\frac{1}{\sqrt{2}})$$ m \ge 1$$ $$n {3}$$ S^{n-1}(\frac{1}{\sqrt{2}}) S^n$$ First, determine sufficient condition ensures nonnegatively defined on $$\mathcal {C}\big ({\tilde{\varphi }}^{-1}T\mathbb S^{n-1}(\frac{1}{\sqrt{2}})\big )$$ Then, complete type analysis our and, as complementary result, obtain p-harmonic final section, compare results those can be deduced from equivariant variation.