Dynamics at infinity and a Hopf bifurcation for a Sprott E system with a very small perturbation constant are studied in this paper. By using Poincaré compactification of polynomial vector fields in $R^{3}$, the dynamics near infinity of the singularities is obtained. Furthermore, in accordance with the centre manifold theorem, the subcritical Hopf bifurcation is analysed and obtained. Numerical simulations demonstrate the correctness of the dynamical and bifurcation analyses. Moreover, by choosing appropriate parameters, this perturbed system can exhibit chaotic,quasiperiodic and periodic dynamics, as well as some coexisting attractors, such as a chaotic attractor coexisting with a periodic attractor for $a > 0$, and a chaotic attractor coexisting with a quasiperiodic attractor for $a = 0$. Coexisting attractors are not associated with an unstable equilibrium and thus often go undiscovered because they may occur in a small region of parameter space, with a small basin of attraction in the space of initial conditions.

Hidden hyperchaotic attractors can be generated with three positive Lyapunov exponents in the proposed 5D hyperchaotic Burke–Shaw system with only one stable equilibrium. To the best of our knowledge, this feature has rarely been previously reported in any other higher-dimensional systems. Unidirectional linear error feedback coupling scheme is used to achieve hyperchaos synchronisation, which will be estimated by using two indicators: the normalised average root-mean squared synchronisation error and the maximum cross-correlation coefficient. The 5D hyperchaotic system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integration. In addition, fractional-order hidden hyperchaotic system will be considered from the following three aspects: stability, bifurcation analysis and FPGA implementation. Such implementations in real time represent hidden hyperchaotic attractors with important consequences for engineering applications.