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Hopf Bifurcation, Positively Invariant Set, and Physical Realization of a New Four-Dimensional Hyperchaotic Financial System
- Source :
- Mathematical Problems in Engineering, Vol 2017 (2017)
- Publication Year :
- 2017
- Publisher :
- Hindawi Limited, 2017.
-
Abstract
- This paper introduces a new four-dimensional hyperchaotic financial system on the basis of an established three-dimensional nonlinear financial system and a dynamic model by adding a controller term to consider the effect of control on the system. In terms of the proposed financial system, the sufficient conditions for nonexistence of chaotic and hyperchaotic behaviors are derived theoretically. Then, the solutions of equilibria are obtained. For each equilibrium, its stability and existence of Hopf bifurcation are validated. Based on corresponding first Lyapunov coefficient of each equilibrium, the analytical proof of the existence of periodic solutions is given. The ultimate bound and positively invariant set for the financial system are obtained and estimated. There exists a stable periodic solution obtained near the unstable equilibrium point. Finally, the dynamic behaviors of the new system are explored from theoretical analysis by using the bifurcation diagrams and phase portraits. Moreover, the hyperchaotic financial system has been simulated using a specially designed electronic circuit and viewed on an oscilloscope, thereby confirming the results of the numerical integrations and its real contribution to engineering.
- Subjects :
- Lyapunov function
Hopf bifurcation
Article Subject
Phase portrait
lcsh:Mathematics
General Mathematics
General Engineering
Chaotic
Financial system
Invariant (physics)
lcsh:QA1-939
01 natural sciences
010305 fluids & plasmas
Nonlinear Sciences::Chaotic Dynamics
symbols.namesake
Nonlinear system
lcsh:TA1-2040
0103 physical sciences
symbols
lcsh:Engineering (General). Civil engineering (General)
010301 acoustics
Bifurcation
Mathematics
Electronic circuit
Subjects
Details
- ISSN :
- 15635147 and 1024123X
- Volume :
- 2017
- Database :
- OpenAIRE
- Journal :
- Mathematical Problems in Engineering
- Accession number :
- edsair.doi.dedup.....8e571e401dab8ffa36254092bca5c204