Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTransient dynamics of the Lorenz system with a parameter drift
73
0
0.0
(
0
)
Added by
Julia Cantis\\'an G\\'omez
Publication date
2020
fields
Physics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Non-autonomous dynamical systems help us to understand the implications of real systems which are in contact with their environment as it actually occurs in nature. Here, we focus on systems where a parameter changes with time at small but non-negligible rates before settling at a stable value, by using the Lorenz system for illustration. This kind of systems commonly show a long-term transient dynamics previous to a sudden transition to a steady state. This can be explained by the crossing of a bifurcation in the associated frozen-in system. We surprisingly uncover a scaling law relating the duration of the transient to the rate of change of the parameter for a case where a chaotic attractor is involved. Additionally, we analyze the viability of recovering the transient dynamics by reversing the parameter to its original value, as an alternative to the control theory for systems with parameter drifts. We obtain the relationship between the paramater change rate and the number of trajectories that tip back to the initial attractor corresponding to the transient state.
rate research
Read More
External and internal factors may cause a systems parameter to vary with time before it stabilizes. This drift induces a regime shift when the parameter crosses a bifurcation. Here, we study the case of an infinite dimensional system: a time-delayed oscillator whose time delay varies at a small but non-negligible rate. Our research shows that due to this parameter drift, trajectories from a chaotic attractor tip to other states with a certain probability. This causes the appearance of the phenomenon of transient chaos. By using an ensemble approach, we find a gamma distribution of transient lifetimes, unlike in other non-delayed systems where normal distributions have been found to govern the process. Furthermore, we analyze how the parameter change rate influences the tipping probability, and we derive a scaling law relating the parameter value for which the tipping takes place and the lifetime of the transient chaos with the parameter change rate.
A three-component dynamic system with influence of pumping and nonlinear dissipation describing a quantum cavity electrodynamic device is studied. Different dynamical regimes are investigated in terms of divergent trajectories approaches and fractal statistics. It has been shown, that in such a system stable and unstable dissipative structures type of limit cycles can be formed with variation of pumping and nonlinear dissipation rate. Transitions to chaotic regime and the corresponding chaotic attractor are studied in details.
Transient chaos is a characteristic behavior in nonlinear dynamics where trajectories in a certain region of phase space behave chaotically for a while, before escaping to an external attractor. In some situations the escapes are highly undesirable, so that it would be necessary to avoid such a situation. In this paper we apply a control method known as partial control that allows one to prevent the escapes of the trajectories to the external attractors, keeping the trajectories in the chaotic region forever. To illustrate how the method works, we have chosen the Lorenz system for a choice of parameters where transient chaos appears, as a paradigmatic example in nonlinear dynamics. We analyze three quite different ways to implement the method. First, we apply this method by building a 1D map using the successive maxima of one of the variables. Next, we implement it by building a 2D map through a Poincar{e} section. Finally, we built a 3D map, which has the advantage of using a fixed time interval between application of the control, which can be useful for practical applications.
In the theoretical research of Hepatitis B virus, mathematical models of its transmission mechanism have been thoroughly investigated, while the dynamics of the immune process in vivo have not. At present, nearly all the existing models are based on integer-order differential equations. However, models restricted to integer-order cannot depict the complex dynamical behaviors of the virus. Since fractional-order models possess property of memory and memory is the main feature of immune response, we propose a fractional-order model of Hepatitis B with time-delay to further explore the dynamical characters of the HBV model. First, by using the Caputo fractional differential and its properties, we obtain the existence and uniqueness of the solution. Then by utilizing stability analysis of fractional-order system, the stability results of the disease-free equilibrium and the epidemic equilibrium are studied according to the value of the basic reproduction number. Moreover, the bifurcation diagram, the largest Lyapunov exponent diagram, phase diagram, Poincare section and frequency spectrum are employed to confirm the chaotic characteristics of the fractional HBV model and to figure out the effects of time-delay and fractional order. Further, a theory of the asymptotical stability of nonlinear autonomous system with delay on the basis of Caputo fractional differential is proved. The results of our work illustrates the rich dynamics of fractional HBV model with time-delay, and can provide theoretical guidance to virus dynamics study and clinical practice to some extent.
Unstable periodic orbits (UPOs) are a valuable tool for studying chaotic dynamical systems. They allow one to extract information from a system and to distill its dynamical structure. We consider here the Lorenz 1963 model with the classic parameters value and decompose its dynamics in terms of UPOs. We investigate how a chaotic orbit can be approximated in terms of UPOs. At each instant, we rank the UPOs according to their proximity to the position of the orbit in the phase space. We study this process from two different perspectives. First, we find that, somewhat unexpectedly, longer period UPOs overwhelmingly provide the best local approximation to the trajectory, even if our UPO-detecting algorithm severely undersamples them. Second, we construct a finite-state Markov chain by studying the scattering of the forward trajectory between the neighbourhood of the various UPOs. Each UPO and its neighbourhood are taken as a possible state of the system. We then study the transitions between the different states. Through the analysis of the subdominant eigenvectors of the corresponding stochastic matrix we provide a novel interpretation of the mixing processes occurring in the system by taking advantage of the concept of quasi-invariant sets.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
