Generation Of Limit Cycles Research Articles

Density-dependent Prenatal Androgen Exposure as an Endogenous Mechanism for the Generation of Cycles in Small Mammal Populations

Small mammal populations exhibit cyclic fluctuations in their population densities. Several hypotheses regarding the mechanisms underlying these population cycles have been advanced, but none has yet gained general approval. We propose here an endogenous mechanism based on the masculinization of female offspring in response to increased population levels. High population levels trigger the non-specific stress response resulting in high levels of circulating androgens in individuals of the population, including pregnant females. These androgens masculinize female offspringin utero, thereby reducing the reproductive capacity of the next generation and subsequently the population size. We have developed and analysed a mathematical model to investigate the possible role of prenatal androgen exposure in the generation of limit cycles. We find the locus of Hopf bifurcations for this model and show that limit cycles depend on three parameters: (1) the delay between birth and sexual maturation; (2) the slope of the function that relates average prenatal androgen exposure to total population density; and (3) the difference between the maximum birth rates of the low- and high-androgen exposed females. We derive the analytical form relating these parameters at the Hopf-bifurcation locus and discuss its biological ramifications. In brief, in each of these three parameters is sufficiently large, population cycles will results from the endogenous mechanism proposed.

Self-organization of temporal structures in a multiequilibrium self-excited oscillator system with frequency control

An analysis is made of the nonlinear dynamics of a model of a self-excited oscillator system with automatic fine tuning of the frequency and possessing more than one equilibrium state. It is shown that, depending on the delay parameters of the control loop and the initial frequency detuning, various periodic and stochastic temporal structures may be formed, accompanied by the generation of various limit cycles and chaotic attractors in phase space. Reasons for the onset of self-modulation are set forth, and the position of the different oscillation regions is established. The main bifurcations are studied together with scenarios for the transformation of the oscillator self-modulation modes as a function of the parameters.

Automatic tuning of decentralized PID controllers for MIMO processes

Abstract An automatic tuning algorithm for decentralized PID control in multiple-input multiple-output (MIMO) plants is presented. This algorithm generalizes the authors' recent auto-tuner for two-input two-output systems to any number of inputs and outputs. The algorithm consists of two stages. In the first, the desired critical point, which consists of the critical gains of all the loops and a critical frequency, is identified. The auto-tuner identifies the desired critical point with almost no a priori information about the process. During the identification phase all controllers are replaced by relays, thus generating limit cycles with the same period in all loops. It is shown that each limit cycle corresponds to a single critical point of the process. By varying the relays parameters different points can be determined. The auto-tuner contains a procedure which converges rapidly to the desired critical point while maintaining the amplitudes of the process variables as well as of the manipulated variables within prespecified ranges. In the second stage, the data of the desired critical point is used to tune the PID controllers by the Ziegler-Nichols rules or their modifications. This paper focuses on the first stage. The steady-state process gains, which are required for the appropriate choice of the desired critical point, are determined by the auto-tuner in closed-loop fashion simultaneously with the identification of the critical point. The identification of the process gains is achieved at no extra plant time. Based upon a large number of simulated cases, the proposed auto-tuner seems to be efficient and robust. The paper discusses the underlying principles of the auto-tuner and its properties and capabilities are demonstrated via examples.

Generation of stable limit cycles in controllable linear systems

In this paper we prove that any controllable linear systems x ˙ = A x + B u , x ∈ ℝ n , u ∈ ℝ m , admits a polynomial feedback u= u( x) such that the closed-loop system x ˙ = A x + B u ( x ) admits an orbitally asymptotically stable limit cycle. Moreover, we prove that for any positive integer n, there exists an nth-order polynomial, autonomous, ordinary differential equation with a unique limit cycle.

Variable Structure Control of Systems with Nonlinear Friction and Dynamic Backlash

This paper investigates the control of systems with nonlinear friction and dynamic backlash. The study is based on the describing function (DF) of nonlinear systems. The controllers studied are the first order model and the second order model variable structure controllers (FOM-VSC and SOM-VSC). The nonlinear plants consist on a mass with kinetic plus viscous friction or two masses subjected to backlash. The worst case for the limit cycle generation is found to be the closed-loop system having two masses with backlash. On the other hand, the closed-loop system of a mass with friction is found to be amenable to control without limit cycles.

Automatic tuning of decentralized PID controllers for TITO processes

This paper presents a new algorithm for automatic tuning of decentralized PID control for two-input two-output (TITO) plants that fully extends the single-loop relay auto-tuner to the multiloop case. The tuning procedure consists of two stages. In the first, the desired critical point, which consists of the critical gains of the two loops and a critical frequency, is identified. Unlike SISO plants, there are infinitely many such points, and knowledge of the desired one is essential to the tuning procedure. The auto-tuner identifies the desired critical point with almost no a priori information on the process. During the identification phase, all controllers are replaced by relays, thus generating limit cycles with the same period in both loops. It is shown that each limit cycle corresponds to a single critical point of the process. By varying the relay parameters, different critical points can be determined. The auto-tuner contains a procedure that converges rapidly to the desired critical point while maintaining the amplitudes of the process variables as well as of the manipulated variables within prespecified ranges. In the second stage, the data of the desired critical point and possibly of other critical points is used to tune the PID controller by the Ziegler-Nichols rule or its modifications. This paper focuses on the first stage. The steady-state process gains, which are required for the appropriate choice of the desired critical point, are determined by the auto-tuner in closed-loop fashion simultaneously with the identification of the critical points. The identification of the process gains is achieved at no extra plant time. On the basis of a large number of simulated cases, the proposed auto-tuner seems to be efficient and robust. This paper discusses the underlying principles of the auto-tuner and its properties and capabilities demonstrated via examples. The algorithm is not limited to TITO cases, and can be extended to any number of loops.

Automatic tuning of PID controllers based on transfer function estimation

A method for the automatic tuning of PID controllers in a closed loop, based on the estimation of a parametric ‘black-box’ transfer function model, is proposed. The system is excited by generating limit cycle oscillations at two different frequencies, which are approximately the crossover frequency and the critical frequency for the feedback loop. A discrete parametric transfer function model is estimated from the experimental data. Important parameters concerning the estimation, such as the prefilter cut-off frequency and the sampling interval, are determined automatically from the experimental data. The PID parameters are determined from a constrained optimization in the frequency domain. The constraints are classical control system properties, such as the maximum amplitudes of the sensitivity and the complementary sensitivity functions. Given these constraints, the PID parameters are determined such that the low frequency amplitude characteristic for the controller is maximized. Simulation experiments show that the tuning procedure has low sensitivity to disturbances and noise during the tuning experiment.

Generation of associative processes in a neural network with realistic features of architecture and units.

A recent neural network model of cortical associative memory incorporating neuronal adaptation by a simplified description of its underlying ionic mechanisms is extended towards more realistic network units and architecture. Excitatory units correspond to groups of adapting pyramidal neurons and inhibitory units to groups of nonadapting interneurons. The network architecture is formed from pairs of one pyramidal and one interneuron unit each with inhibitory connections within and excitatory connections between pairs. The degree of adaptability of the pyramidal units controls the character of the network dynamics. An intermediate adaptability generates limit cycles of transitions between stored patterns and regulates oscillation frequencies in the range of theta rhythms observed in the brain. In particular, neuronal adaptation can impose a direction of transitions between overlapping patterns also in a symmetrically connected network. The model permits a detailed analysis of the transition mechanisms. Temporal sequences of patterns thus formed may constitute parts of associative processes, such as recall of stored sequences or search of pattern subspaces. As a special case, neuronal adaptation can accomplish pattern segmentation by which overlapping patterns are temporally resolved. The type of limit cycles produced by neuronal adaptation may also be of significance for central pattern generators, also for networks involving motor neurons. The applied learning rule of Hebbian type is compared to a modified version also common in neural network modelling. It is also shown that the dependence of the network dynamic behaviour on neuronal adaptability, from fixed point attractors at weak adaptability towards more complex dynamics of limit cycles and chaos at strong adaptability, agrees with that recently observed in a more abstract version of the model. The present description of neuronal adaptation is compared to models based on dynamic firing thresholds.

Automatic Tuning of Decentralized PID Controllers for TITO Processes

The paper presents a new automatic tuning algorithm for Decentralized PID Control for two inputs - two outputs (TITO) plants. The tuning procedure, which is an extension of the Ziegler-Nichols method requires identification of a critical point consisting of critical gains of the various loops and a critical frequency. Unlike SISO plants there are infinitely many such points and knowledge of the desired one is essential to the tuning procedure. The auto-tuner identifies the desired critical point with no apriori information of the process. During the identifiction phase all controllers are replaced by relays, thus generating limit cycles with the same period in both loops. It is shown that this limit cycle corresponds to a single critical point of the process. By varying the relays parameters different critical point can be determined. The auto-tuner contains a procedure which converges rapidly to the desired critical point while maintaining the amplitudes of the process variables within prespecified ranges. The steady-state process gains, which are required for the appropriate choice of the desired critical point, are determined in closed-loop fashion simultaneously with the identification of the critical points. It is shown that the proposed auto-tuner is more efficient and reliable than a recently published algorithm and works well also in cases where the latter algorithm fails to converge and even destabilizes the system

Small Amplitude Limit Cycles for Cubic Systems

AbstractIn this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systemsWith a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.

On internal resonance of nonlinear, vibrating systems with many degrees of freedom

The problem of periodic solutions of nonlinear autonomous systems with many degrees of freedom is considered. This is made possible by the development of a modified version of the KBM method[1]. The method can be used to generate limit cycle phase portrait, amplitude, period and to indicate stability of the limit cycle.

Perturbations of a Hamiltonian family of cubic vector fields

This paper is related with the configurations of limit cycles for cubic polynomial vector fields in two variables (χ3).It is an open question to decide whether every limit cycle configuration in χ3 can be obtained by perturbation of a corresponding Hamiltonian configuration of centres and graphs.In this work, by considering perturbations of the Hamiltonian vector field XH = (Hy, − Hx), where H(x, y) = [a(x + h)2 + by2 − 1] [a(x − h)2 + by2 − 1], we make a global analysis of the possible cases.The vector field XH has three centres (C−, C+ and the origin) and two saddles. By means of quadratic perturbations the centres become fine foci where C−and C+ have the same type of stability but opposed to that one of the origin and infinity. Further introducing cubic perturbations changes the stability of C−, C+ and the cycle at infinity and generates limit cycles. Lastly extra linear terms change the stability of the origin and generate another limit cycle.Finally, we analyse the rupture of saddle connection of the Hamiltonian field under perturbation, via Melnikov's integral, in order to complete the study of the global phase portrait and to consider the possibility of new limit cycles emerging from the Hamiltonian graph.

Condition for generating limit cycles by bifurcation of the loop of a saddle-point separatrix

In paper [1] (On the stability of a saddle-point separatrix loop and analytical criterion for its bifurcation limit cycles Acta Mathematica Sinica Vol. 28. No. 1, 55–70, Bejing China 1985), we considered the problem of generating limit cycles by the bifurcation of a stable or an unstable loop of a saddle-point separatrix. We gave for the first time a criterion for the stability of the loop as following:L0 is stable (unstable) if\(\int_{ - \infty }^\infty {(P'_{0x} + Q'_{0y} )dt 0)} \) wherex=ϕ(t),y=ϕ(t) then a sufficient condition for the bifurcation which generates limit cycles. This paper generalizes the result of [1] to the case where the loop contains a center or the loop tends to an infinite saddle-point, and removes the restriction that the saddle-point should be an elementary singular point. Applying the results of this paper, the author studies a two-parameter system $$\left\{ \begin{gathered} x = lx^2 + y^2 - y + 5\varepsilon xy \hfill \\ y = (3l + 5)xy + x + \varepsilon x^2 \hfill \\ \end{gathered} \right.$$

Generation of multiple attractors and nonequilibrium phase transitions

We develop and discuss the generation of new limit cycles in chemical systems by external periodic perturbations. The deterministic analysis is based on prior work by Loud. We choose a model derived from experiments, the conversion of 2,3 epoxy-1-propanol to glycerine. The autonomous system of equations which govern the dynamics of this system has a limit cycle arising from a normal Hopf bifurcation, and a stable focus, and a limit cycle associated with an inverted Hopf bifuraction. For specific choices of periodic perturbations a prescribed number of periodic attractors can be generated from a single periodic attractor (normal Hopf bifurcation); and two periodic attractors, a periodic and biperiodic, or two biperiodic attractors can be generated from a stable focus and periodic attractor (inverted Hopf bifurcation). Noise is then imposed on the attractors of the autonomous system and the multiple attractors of the externally forced system. Noise imposed on the autonomous system with a stable focus and limit cycle causes random transitions, analogous to first order phase transitions. This behavior has some resemblance to intermittency noted in some biological systems. From the results of noise imposed on systems with different numbers of isoperiodic limit cycles we suggest, after a limited number of observations, the trend that as the number of attractors increases, the lines in the power spectrum narrow and the area in the phase plane within contours of equal probability increases.

Sensitivity to switching-function variations in a time-optimal positional system

The paper deals with the sensitivity to switching-function variations of a time–optimal system which is controlling a positional mechanism. The model chosen has a second-order differential equation with non-linear and discontinuous right–hand side. Such a model covers a large class of motion resistances, in particular all types of friction. Differential inequality theory and the theory of discontinuous differential equations, and in particular a method of generalized jT solutions, are used to show that time-optimal switching-function variations cause the origin to become an unstable equilibrium point of the state-plane and the control system to generate limit–cycles. Some suggestions for applications are given, as is a practical example.

Non-linear oscillator limit cycle analysis using a time transformation approach

A method is presented for the analysis of limit cycle behavior of autonomous non-linear oscillators characterized by second order ordinary differential equations containing a small parameter. The method differs from the classical perturbation methods in that the dependent variable is not expanded in a power series in the small parameter. Rather, a new independent variable is sought such that in its domain the motion is simple harmonic. Use of this time transformation technique to generate limit cycle phase portrait, amplitude and period is presented. We show results of the application of the method to the van der Pol oscillator, to an oscillator with quadratic damping, and to a modified van der Pol oscillator which is statically unstable in the limit of small motion.

A theoretical study of limit cycle oscillations of plenum air cushions

Air cushion vehicles (ACV) are prone to the occurrence of dynamic instabilities which frequently appear as stable finite amplitude oscillations. The aim of this work is to ascertain if the non-linearities characteristics of ACV dynamics generate limit cycle oscillations for cushion systems operating at conditions for which a linear theory predicts instability. The types of non-linearity that can occur are discussed, and an analysis is presented for a single cell flexible skirted plenum chamber constrained to move in pure heave only. Two cushion feed cases are considered: a plenum box supply and a duct. The results obtained by a Galerkin/describing function analysis are compared with those generated by a full numerical simulation. For the plenum box supply system, it is shown that the limit cycles can be suppressed by using a piston to introduce high frequency small amplitude volume oscillations into the plenum chamber.

Spherical drop of cytoplasm with an effective surface tension influenced by oscillating enzymatic reactions. A simplified model for the biological clock, cell division, and chemotaxis

A model suggested as a possible mechanism of a biological clock of the escapement type is constructed. A biological cell is treated as a spherical drop of cytoplasm with embedded contractile elements (microfilaments and microtubules). The contractile elements are thought to be positioned mainly in the thin layer of ectoplasm adjacent to the cell membrane. This layer is described as a mathematical surface with an “effective surface tension” and an “effective surface viscosity.” For simplicity, the synthesis of contractile elements is thought to be effectuated by one or more of the enzymes in a triangular enzyme cycle (Christiansen cycle). The enzyme concentrations in this model may perform damped oscillations around their stationary values, just as enzymes in real enzyme cycles. A fluid drop with surface tension is also able to perform damped oscillations. When the two systems are combined, calculations show that unstable, oscillatory solutions may be obtained. Those may generate limit cycles to be used in a mechanochemical clock. In a cell tissue we have the possibility of mechanical interaction between the cells and of nonlinear synchronization of the individual clocks. If there should not be a complete decoupling between the hydrodynamic and the chemical dispersion relations between complex growth constant and normal mode number, it is very important that the three surface diffusion coefficients of the enzymes are not all equal. Apart from that, calculations reveal several remarkable features of the model: (a) Stationary instabilities have loci which are independent of the absolute values of surface tension. The oscillatory modes have loci of marginal stability which depend on surface tension, except for the l = 1 surface harmonics. (b) For l = 1 (translation mode), stationary instability appears before oscillatory instability. This mode may therefore be used to model chemotaxis of unicellular organisms in the gradient of an external attractant. (c) The mode l = 2 (cell division mode) exhibits oscillatory instability before the appearance of stationary instability when the coupling parameter between enzyme kinetics and hydrodynamics is increased. The model may therefore be used to simulate the mitotic clock as well as the cell division process and various disturbances of the threshold values leading to cancer growth. (d) With the present model it is easy to obtain a perfect compensation of the temperature dependence of the chemical rate constants.

Time domain and realization of a class of digital hybrid filter

AbstractThis paper describes a time‐domain design and realization of a digital hybrid filter with a specific cutoff frequency. The proposed realization is obtained by adding a single multiplier in a feedback loop to the low‐pass or high‐pass FIR filters with linear phase. The resulting hybrid filters can have the following improved frequency response: (1) Increased attenuation in the stop‐band (compared to that obtainable with a Kaiser windowed FIR filter of the same order) for the same transition bandwidth; (2) almost constant amplitude response in the stopband and no zero‐cross ripple; (3) no degradation of phase linearity in the pass‐band; and (4) almost constant phase response in the stopband. Also the hybrid filter designed and realized in this way is always stable and does not generate limit cycles. The time‐domain design and realization of digital hybrid filters of this class are introduced and a design example of a low‐pass hybrid filter is given.

Evolutionary Aspects of Demographic Cycles: The Relevance of Some Models of Cycles for Microtine Fluctuations

Evolutionary aspects of some current theories for regular density oscillations are discussed. General problems in testing fitness differences are briefly discussed. It is pointed out that two different interpretations of Chitty's theory are common in literature: One states that regular density oscillations are caused by genetically based behavioural polymorphism in the absence of extrinsic changes; the other that regular cycles are the result of the interaction between genetic changes and extrinsic factors. Only the latter is concluded plausible. However, further empirical research attempting to test the hypothesized lower general fitness of the aggressive strategies is needed. Genetic changes in parasites may, on the basis of preliminary theoretical studies, be presumed likely to result in cycles. More experimental work is needed before any conclusions can be drawn. Genetic feedback interactions between predator-prey interactions are likely to be rejected as unimportant for generating limit cycles. Exceptions may be predator-prey interactions in simple habitat complexes and possibly in the optimal area of the prey's distribution range. Female biased sex ratios is discussed: It is concluded that female biased sex ratios are likely to be a result of both evolutionary and ecological forces. It is suggested that the study of skewed sex ratios may be rewarding for the study of microtine cycles.