Global Stability of a Markovian Jumping Chaotic Financial System with Partially Unknown Transition Rates under Impulsive Control Involved in the Positive Interest Rate

Abstract

The intrinsic instability of the financial system itself results in chaos and unpredictable economic behavior. To gain the globally asymptotic stability of the equilibrium point with a positive interest rate of the chaotic financial system, pulse control is sometimes very necessary and is employed in this paper to derive the globally exponential stability of financial system. It should be pointed out that the delayed feedback model brings an essential difficulty so that the regional control method has to be adopted. In this paper, the author firstly employs impulsive control, regional control, the Lyapunov function technique, and variational methods to derive the stochastically globally asymptotic stability criterion of the economic balance point with a positive interest rate for a delayed feedback financial system with Markovian jumping and partially unknown transition rates. Besides, the mathematical induction method and the proof by contradiction are applied synthetically to deduce the globally exponential stability of the equilibrium point with a positive interest rate for the impulsive financial system without time-delays. Moreover, numerical examples illustrate that under suitable data conditions on the two main criteria mentioned above, the interest rates are positive decimals when the financial system reaches stability, which means better economic significance.

Input-to-state stability and no-inputs stabilization of delayed feedback chaotic financial system involved in open and closed economy

“…Therefore, no-inputs stabilization of delayed feedback chaotic financial system is considered in this paper, too. Specifically, we are interested in the following financial systems that is composed of the production sub-block, currency sub-block, securities sub-block, and labor sub-block (see, e.g., [3,5,7,18,19,23,35,36]):…”

“…where x represents the interest rate, y represents the investment demand, z represents the price index, a represents savings, b represents the unit investment cost, and c represents the elasticity of commodity demand. It is well known ( [18,19,23]) that the financial system (1) has the unique equilibrium point Q 0 (0, 1 b , 0) if c − b − abc ≤ 0, and owns three equilibrium point Q 0 (0, 1 . Under some suitable data, the equilibrium point Q 0 (0, 1 b , 0) may be stable.…”

“…Remark that both the interest rates of Q 0 and Q 2 are non-positive. Moveover, according to the principles of Macroeconomics, the interest rate is usually a positive decimal whenever the product market and the money market are in common equilibrium ( [23]). So, in this paper, only the stability of the equilibrium point Q 1 will be studied.…”

Impulsive control on delayed feedback chaotic financial system with Markovian jumping

“…Moreover, global exponential stability implies deleting chaos of complex economy system. As pointed out in [8] and [30], under Lipschitz conditions ensuring the unique existence of the solution of the reaction-diffusion system for any given initial value, Ruofeng Rao, Shouming Zhong, and Zhilin Pu deduced the boundedness conclusion [30,Theorem 3.3] and the stability criterion [30,Theorem 3.4], in which the following formula was derived: This has actually proven the following conclusion.…”

pth Moment Stability of a Stationary Solution for a Reaction Diffusion System with Distributed Delays

“…However, in practical engineering, the time delay is unavoidable, which may lead to chaos and instability of the system [16][17][18][19][20][21][22][23][24]. Thus, in this paper, we are to investigate the delayed p-Laplacian reaction-diffusion dynamic system.…”