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In this work some complex behaviors of a controlled reverse flow reactor is presented. The control system introduces discrete events making the model an infinite dimensional hybrid system. The study is conducted through continuation techniques and brute force numerical simulations. Together with standard bifurcations like pitchfork, saddle-node and Neimark–Sacker, varying the set-point parameter of the controller, several novel aspects are singled out: an unusual sequence of period-adding bifurcation phenomena, a new route to chaos and the coexistence of Zeno states with quasi-periodic and chaotic regimes. The period-adding phenomena dictate the transition between symmetric and asymmetric multiperiodic regimes and a simple rule for the occurrence of symmetry breaking and recovery is found. The new route to chaos is a transition from a quasi-periodic regime to chaos due to the presence of Zeno phenomena, typical of hybrid systems. The chaos is characterized by Zeno-like oscillations.

In this paper, some complex phenomena of dynamical bifurcations are shown for a two-neural network with delay coupling. The sigmoid activation function with slope ratio, a monotonically increasing function, is proposed to consider the relations of the sigmoid and Hardlim functions. The equilibrium points are studied analytically in detail in terms of the characteristic equation and static bifurcation. The central manifold reduction and normal form method are employed to determine Hopf bifurcation and its stability. The stable equilibrium points and periodic motions are observed in different parameter regions. Effects of slope ratio and delayed coupling on dynamic behaviors are investigated by the numerical simulation, such as Poincare map, phase portraits, and power spectrum. Various active transitions to chaos and the corresponding critical boundaries on the focused parameter regions are obtained to classify dynamical behaviors including stable equilibrium point, periodic solution, 2-torus, 3-torus, and then the chaotic motions.

In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of $2.15$. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

Starting from historical researches, we used, like Van der Pol and Le Corbeiller, a cubic function for modeling the current–voltage characteristic of a direct current low-pressure plasma discharge tube, i.e. a neon tube. This led us to propose a new four-dimensional autonomous dynamical system allowing to describe the experimentally observed phenomenon. Then, mathematical analysis and detailed numerical investigations of such a fourth-order torus circuit enabled to highlight bifurcation routes from torus breakdown to homoclinic chaos following the Newhouse–Ruelle–Takens scenario.

We consider the dynamical effects of electromagnetic flux on the discrete Chialvo neuron model. It is shown that the model can exhibit rich dynamical behaviors such as multistability, firing patterns, antimonotonicity, closed invariant curves, various routes to chaos, and fingered chaotic attractors. The system enters a chaos regime via period-doubling cascades, reverse period-doubling route, antimonotonicity, and via a closed invariant curve to chaos. The results were confirmed using the techniques of bifurcation diagrams, Lyapunov exponent diagram, phase portraits, basins of attraction, and numerical continuation of bifurcations. Different global bifurcations are also shown to exist via numerical continuation. After understanding a single neuron model, a network of Chialvo neurons is explored. A ring-star network of Chialvo neurons is considered and different dynamical regimes such as synchronous, asynchronous, and chimera states are revealed. Different continuous and piecewise continuous wavy patterns were also found during the simulations for negative coupling strengths.

Chaos is ubiquitous in nonlinear dynamical systems. It is a hidden state of deterministic disorder in certain regions of parameters and initial conditions of a nonlinear system. The path taken by the system to reach chaos is known as its route to chaos. The state of the system can drastically change from a stable fixed point or a periodic solution to chaos due to the effect of local or global bifurcation. The available tools, such as bifurcation diagrams, Lyapunov spectra, Poincaré maps, phase plots, and time series plots, explore the routes for such qualitative changes. These are tiresome procedures. For more efficient verifications, this work offers a guide or catalog for several routes to chaos (RTC) through which a continuous-time dynamical system enters or exits from chaos. Additionally, this review paper highlights various signatures associated with common RTC to make identification easier. The main emphasis is on enabling a reader to assess these RTC by merely observing the explained signatures in the bifurcation diagrams and the Poincaré maps.