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The rapid development of Deep Neural Networks (DNNs) lays solid foundations for Internet of Things systems. However, mobile devices with limited processing capacity and short battery life confront the difficulties of executing complex DNNs. To satisfy different Quality of Service requirements, a feasible solution is offloading DNN layers to edge nodes and the cloud. The energy-efficient offloading problem for DNN-based applications with the deadline and budget constraints in the edge-cloud environment is still an open and challenging issue. To this end, this paper proposes a Hybrid Chaotic Evolutionary Algorithm (HCEA) incorporating diversification and intensification strategies and a DVFS-enabled version of it (HCEA-DVFS). The Archimedes Optimization Algorithm-based diversification strategy exploits global and local guiding information to improve population diversity during the updating process and employs Metropolis acceptance rule of Simulated Annealing to avoid premature convergence. The Genetic Algorithm-based chaotic intensification strategy is designed to enhance the local search capability of HCEA. Moreover, the Dynamic Voltage Frequency Scaling-enabled adjustment strategies can be embedded into HCEA to further reduce energy consumption by resetting frequency levels and reallocating DNN layers. Experimental results over four DNN-based applications demonstrate that HCEA-DVFS can reduce more energy consumption under different deadlines, budgets, and workloads on average by 7.93, 9.68, 11.02, 11.84, and 19.38 percent in comparison with HCEA, PSO-GA, MCEA, AOA, and Greedy, respectively. • An AOA-based diversification strategy is designed to exploit global and local guiding information for population updating. • A chaotic GA-based intensification strategy is developed to enhance local search capability and introduce randomness. • A DVFS-enabled adjustment strategy is presented to save energy by resetting frequency levels and reallocating DNN layers. • HCEA and HCEA-DVFS outperform baselines in energy consumption optimization under budget and deadline constraints. [ABSTRACT FROM AUTHOR]