In this paper, we proposed a class of two-dimensional three-parameter discrete dynamical systems with cubic terms, which can be applied to image encryption. We present a study on the analysis and control of the bifurcations in these systems. Initially, the existence and stability conditions of the fixed points of the proposed systems were proposed. Subsequently, based on the center manifold and bifurcation theories, we determined the conditions for the existence of Neimark-Sacker, pitchfork, and period-doubling bifurcations. The bifurcation diagram and the phase portraits were employed in the numerical experiments to verify the correctness of theoretical analysis. Finally, anti-controllers were used to induce Neimark-Sacker and period-doubling bifurcations, which were designed by integrating the bifurcation conditions with the state feedback method. The proposed anti-controllers caused the systems to undergo the desired bifurcations at the preset parameter values. Numerical simulations verified the effectiveness and robustness of the proposed controllers.

In this paper, we studied the continuous dependence and stability of solutions for a class of fractional partial differential equations with multiple spatially varying coefficient parameters. The nonlocal operator was defined by a symmetric kernel, yielding a self-adjoint structure essential to the analysis. Using variational methods and the Minty–Browder theorem, we established the existence and uniqueness of weak solutions in the energy space $X_0$ for each admissible parameter vector $w$. We extended single-parameter stability to a multi-parameter framework by proving that the solution operator $S_f$ is continuous with respect to $w$ in the product space $\prod _i{L}^{q_i}(\mathrm{\Omega })$. Moreover, we derived an explicit global Lipschitz estimate for $S_f$ and showed its Gâteaux differentiability under mild regularity assumptions on $f(x,u,w)$. Numerical simulations confirmed continuity, Lipschitz stability, and differentiability of $S_f$ with respect to all parameters. These results provided rigorous guarantees for inverse problems and uncertainty quantification in multi-parameter fractional PDE models.