Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka...Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.展开更多
We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the...We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.展开更多
In this paper the author presents an overview on his own research works.More than ten years ago,we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation.T...In this paper the author presents an overview on his own research works.More than ten years ago,we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation.That is the stochastic velocity type’s Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation.This equation is time-reversed asymmetrical.It shows that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality,and the law of motion of statistical thermodynamics is expressed by a superposition of both the law of dynamics and the stochastic velocity and possesses both determinism and probability.Hence it is different from the law of motion of particles in dynamical systems.The stochastic diffusion motion of the particles is the microscopic origin of macroscopic irreversibility.Starting from this fundamental equation the BBGKY diffusion equation hierarchy,the Boltzmann collision diffusion equation,the hydrodynamic equations such as the mass drift-diffusion equation,the Navier-Stokes equation and the thermal conductivity equation have been derived and presented here.What is more important,we first constructed a nonlinear evolution equation of nonequilibrium entropy density in 6N,6 and 3 dimensional phase space,predicted the existence of entropy diffusion.This entropy evolution equation plays a leading role in nonequilibrium entropy theory,it reveals that the time rate of change of nonequilibrium entropy density originates together from its drift,diffusion and production in space.From this evolution equation,we presented a formula for entropy production rate(i.e.the law of entropy increase)in 6N and 6 dimensional phase space,proved that internal attractive force in nonequilibrium system can result in entropy decrease while internal repulsive force leads to another entropy increase,and derived a common expression for this entropy decrease rate or another entropy increase rate,obtained a theoretical expression for unifying thermodynamic degradation and self-organizing evolution,and revealed that the entropy diffusion mechanism caused the system to approach to equilibrium.As application,we used these entropy formulas in calculating and discussing some actual physical topics in the nonequilibrium and stationary states.All these derivations and results are unified and rigorous from the new fundamental equation without adding any extra new assumption.展开更多
文摘Discrete Lotka-Volterra systems in one dimension (the logistic equation) and two dimensions have been studied extensively, revealing a wealth of complex dynamical regimes. We show that three-dimensional discrete Lotka-Volterra dynamical systems exhibit all of the dynamics of the lower dimensional systems and a great deal more. In fact and in particular, there are dynamical features including analogs of flip bifurcations, Neimark-Sacker bifurcations and chaotic strange attracting sets that are essentially three-dimensional. Among these are new generalizations of Neimark-Sacker bifurcations and novel chaotic strange attractors with distinctive candy cane type shapes. Several of these dynamical are investigated in detail using both analytical and simulation techniques.
基金supported by National Natural Science Foundation of China(Grant Nos.12171492 and 12201135).
文摘We provide an alternative characterization of the invariance entropy for uncertain control systems via covers.We also introduce dimension types of invariance entropies and give a dimension-like characterization of the invariance entropy for uncertain control systems.Moreover,we present four versions of measure-theoretic invariance entropies,give an inverse variational principle of the measure-theoretic Bowen invariance entropy,and obtain four positive variational principles of invariance entropies of subsets with finite elements for uncertain control systems.
文摘In this paper the author presents an overview on his own research works.More than ten years ago,we proposed a new fundamental equation of nonequilibrium statistical physics in place of the present Liouville equation.That is the stochastic velocity type’s Langevin equation in 6N dimensional phase space or its equivalent Liouville diffusion equation.This equation is time-reversed asymmetrical.It shows that the form of motion of particles in statistical thermodynamic systems has the drift-diffusion duality,and the law of motion of statistical thermodynamics is expressed by a superposition of both the law of dynamics and the stochastic velocity and possesses both determinism and probability.Hence it is different from the law of motion of particles in dynamical systems.The stochastic diffusion motion of the particles is the microscopic origin of macroscopic irreversibility.Starting from this fundamental equation the BBGKY diffusion equation hierarchy,the Boltzmann collision diffusion equation,the hydrodynamic equations such as the mass drift-diffusion equation,the Navier-Stokes equation and the thermal conductivity equation have been derived and presented here.What is more important,we first constructed a nonlinear evolution equation of nonequilibrium entropy density in 6N,6 and 3 dimensional phase space,predicted the existence of entropy diffusion.This entropy evolution equation plays a leading role in nonequilibrium entropy theory,it reveals that the time rate of change of nonequilibrium entropy density originates together from its drift,diffusion and production in space.From this evolution equation,we presented a formula for entropy production rate(i.e.the law of entropy increase)in 6N and 6 dimensional phase space,proved that internal attractive force in nonequilibrium system can result in entropy decrease while internal repulsive force leads to another entropy increase,and derived a common expression for this entropy decrease rate or another entropy increase rate,obtained a theoretical expression for unifying thermodynamic degradation and self-organizing evolution,and revealed that the entropy diffusion mechanism caused the system to approach to equilibrium.As application,we used these entropy formulas in calculating and discussing some actual physical topics in the nonequilibrium and stationary states.All these derivations and results are unified and rigorous from the new fundamental equation without adding any extra new assumption.
