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Entropy 2016, 18(10), 371; doi:10.3390/e18100371

Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays

Department of Mathematics and Physics, Bengbu University, Bengbu 233030, China
School of Economics, South-Central University for Nationalities, Wuhan 430074, China
Author to whom correspondence should be addressed.
Academic Editor: J. A. Tenreiro Machado
Received: 2 August 2016 / Revised: 13 October 2016 / Accepted: 13 October 2016 / Published: 19 October 2016
(This article belongs to the Section Complexity)


In this paper, we build a model of energy-savings and emission-reductions with two delays. In this model, it is assumed that the interaction between energy-savings and emission-reduction and that between carbon emissions and economic growth are delayed. We examine the local stability and the existence of a Hopf bifurcation at the equilibrium point of the system. By employing System Complexity Theory, we also analyze the impact of delays and the feedback control on stability and entropy of the system are analyzed from two aspects: single delay and double delays. In numerical simulation section, we test the theoretical analysis by using means bifurcation diagram, the largest Lyapunov exponent diagrams, attractor, time-domain plot, Poincare section plot, power spectrum, entropy diagram, 3-D surface chart and 4-D graph, the simulation results demonstrating that the inappropriate changes of delays and the feedback control will result in instability and fluctuation of carbon emissions. Finally, the bifurcation control is achieved by using the method of variable feedback control. Hence, we conclude that the greater the value of the control parameter, the better the effect of the bifurcation control. The results will provide for the development of energy-saving and emission-reduction policies. View Full-Text
Keywords: energy-savings; emission-reduction; entropy; stability; two delays; Hopf bifurcation energy-savings; emission-reduction; entropy; stability; two delays; Hopf bifurcation

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Wang, J.; Wang, Y. Study on the Stability and Entropy Complexity of an Energy-Saving and Emission-Reduction Model with Two Delays. Entropy 2016, 18, 371.

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