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A new model-free method has been developed and termed the landscape dynamic network biomarker (l -DNB) methodology. The method is based on bifurcation theory, which can identify tipping points prior to serious disease deterioration using only single-sample omics data. Here, we show that l -DNB provides early-warning signals of disease deterioration on a single-sample basis and also detects critical genes or network biomarkers (i.e. DNB members) that promote the transition from normal to disease states. As a case study, l -DNB was used to predict severe influenza symptoms prior to the actual symptomatic appearance in influenza virus infections. The l -DNB approach was then also applied to three tumor disease datasets from the TCGA and was used to detect critical stages prior to tumor deterioration using an individual DNB for each patient. The individual DNBs were further used as individual biomarkers in the analysis of physiological data, which led to the identification of two biomarker types that were surprisingly effective in predicting the prognosis of tumors. The biomarkers can be considered as common biomarkers for cancer, wherein one indicates a poor prognosis and the other indicates a good prognosis. [ABSTRACT FROM AUTHOR]

We classify the local bifurcations of quasi-periodic -dimensional tori in maps (abbr. MT) and in flows (abbr. FT) for . It is convenient to classify these bifurcations into normal bifurcations and resonance bifurcations. Normal bifurcations of MT can be classified into four classes: namely, saddle-node, period doubling, double covering, and Neimark-Sacker bifurcations. Furthermore, normal bifurcations of FT can be classified into three classes: saddle-node, double covering, and Neimark-Sacker bifurcations. These bifurcations are determined by the type of the dominant Lyapunov bundle. Resonance bifurcations are well known as phase locking of quasi-periodic solutions. These bifurcations are classified into two classes for both MT and FT: namely, saddle-node cycle and heteroclinic cycle bifurcations of the -dimensional tori. The former is reversible, while the latter is irreversible. In addition, we propose a method for analyzing higher-dimensional tori, which uses one-dimensional tori in sections (abbr. ST) and zero-dimensional tori in sections (abbr. ST). The bifurcations of ST can be classified into five classes: saddle-node, period doubling, component doubling, double covering, and Neimark-Sacker bifurcations. The bifurcations of ST can be classified into four classes: saddle-node, period doubling, component doubling, and Neimark-Sacker bifurcations. Furthermore, we clarify the relationship between the bifurcations of ST/ST and the bifurcations of MT/FT. We present examples of all of these bifurcations. [ABSTRACT FROM AUTHOR]

The Nagumo-Sato model is a simple mathematical expression of a single neuron, and it is categorized as a discrete-time hybrid dynamical system. To compute bifurcation sets in such a discrete-time hybrid dynamical system accurately, conditions for periodic solutions and bifurcations are formulated herewith as a boundary value problem, and Newton's method is implemented to solve that problem. As the results of the analysis, the following properties are obtained: border-collision bifurcations play a dominant role in dynamical behavior of the model; chaotic regions are distinguished by tangent bifurcations; and multistable attractors are observed in its coupled system. We demonstrate several bifurcation diagrams and corresponding topological properties of periodic solutions. [ABSTRACT FROM AUTHOR]

SUMMARY This paper proposes a method for calculation of local bifurcation points in discrete-time dynamical systems with piecewise nonlinear characteristics (PNDDS). First, an -dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, a calculation method for the local bifurcation points that utilizes the conditional equation for the periodic solution and the characteristic equation is proposed. It is essential to calculate the derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key component of the proposed method, and is explained in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points in order to confirm the validity of the proposed method. [ABSTRACT FROM AUTHOR]

We propose a resilient control scheme to avoid catastrophic transitions associated with saddle-node bifurcations of periodic solutions. The conventional feedback control schemes related to controlling chaos can stabilize unstable periodic orbits embedded in strange attractors or suppress bifurcations such as period-doubling and Neimark-Sacker bifurcations whose periodic orbits continue to exist through the bifurcation processes. However, it is impossible to apply these methods directly to a saddle-node bifurcation since the corresponding periodic orbit disappears after such a bifurcation. In this paper, we define a pseudo periodic orbit which can be obtained using transient behavior right after the saddle-node bifurcation, and utilize it as reference data to compose a control input. We consider a pseudo periodic orbit at a saddle-node bifurcation in the Duffing equations as an example, and show its temporary attraction. Then we demonstrate the suppression control of this bifurcation, and show robustness of the control. As a laboratory experiment, a saddle-node bifurcation of limit cycles in the BVP oscillator is explored. A control input generated by a pseudo periodic orbit can restore a stable limit cycle which disappeared after the saddle-node bifurcation. [ABSTRACT FROM AUTHOR]

We consider a general uncertain nonlinear dynamical system defined in a certain model set, and reformulate a problem of robustness bifurcation analysis (RBA), which has been originally formulated in our previous work. As such, we develop an efficient computational method for the RBA, which can be used for quantitative evaluation of bifurcation robustness in uncertain dynamical systems. Specifically, we first linearize the uncertain system properly and then apply a feedback transformation technique to reduce the RBA problem to a linear robustness analysis one, which can be solved using μ-analysis, a common analysis technique in robust control theory. Finally, we provide robustness analysis of a gene regulatory network model where oscillatory behavior appears according to Hopf bifurcation. We give quantitative evaluation of the bifurcation robustness using the RBA method proposed here. [ABSTRACT FROM AUTHOR]

SUMMARY Bifurcation phenomena in a composite dynamic system are studied in order to understand the basic properties of systems. Recently, it has been reported that unavoidable non-ideal switching, e.g., spike noise or a time delay, occurs due to switching and seriously affects the behavior of the trajectory in a composite dynamical system operating in high-frequency switching ranges. We have analyzed the basic properties of a simple one-dimensional composite dynamical system with nonideal switching in order to understand the essence of the dynamical effects of nonideal switching. In the engineering field, there are many two- or more dimensional systems. Naturally, nonideal switching can occur in two- or more dimensional systems. However, no paper analyzes the effect of nonideal switching in such systems. In this paper, we study the basic properties of a two-dimensional composite dynamical system with spike noise. First, we describe a model of a two-dimensional composite dynamical system. Next, the behavior of the waveforms in a system with ideal switching and a system with spike noise is shown. Then, we sample the data of the waveforms in every period for the external force and define a Poincaré map. Finally, using the Poincaré map, we derive two-parameter bifurcation diagrams and discuss the basic properties of a system with spike noise. [ABSTRACT FROM AUTHOR]

We prove that the active power through a branch (referred to as branch active power) in a power system is a continuous function of the bifurcation parameter in the closed interval from an initial parameter value to the bifurcation-point value of a saddle-node bifurcation (SNB) or limit-induced bifurcation (LIB) of the power flow equation (PFE). Then we show that generally there is a sequence of branch active powers meeting maxima when the parameters are equal to intermediate values in the closed interval. These results can be used to qualitatively evaluate and classify the state of power system operation conditions in terms of voltage stability. Also, they well explain the limitation of the direct current (DC) power flow methods on stressed power systems. Numerical simulations of the IEEE 118-bus and 57-bus systems are used to illustrate the studies and their applications. [ABSTRACT FROM PUBLISHER]

In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincaré section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincaré section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A+ (annulus and orientation preserving), A- (annulus and orientation reversing), M (Möbius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A+ × A+, A- × M (equivalently M × A- and M × M), and F × F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A+ and A- DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark-Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A+ × A+ DLBs correspond to saddle-node bifurcations, type A- × M DLBs to double-covering bifurcations, and type F × F DLBs to the Neimark-Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples. [ABSTRACT FROM AUTHOR]

We study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two-dimensional Hindmarsh-Rose (2DHR) neuron model: the saddle node on an invariant circle (SNIC) bifurcation to the supercritical Andronov-Hopf (AH) one, the SNIC bifurcation to the saddle-separatrix loop (SSL) one, the AH bifurcation to the subcritical AH (SAH) one, and the SSL bifurcation to the AH one. For this purpose, we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2DHR neuron model. These slopes are mathematically formulated with phase response curves (PRCs), expanding the firing rate with perturbations of the temperature and external input current on the one-dimensional space of the phase S1 in the 2DHR oscillator. By analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current, we find that during changes of the firing rate in all of the bifurcation transitions, the calculated slope with respect to the temperature also changes. This is largely dependent on changes in the PRC size that is also related to the slope with respect to the external input current. Furthermore, we find phase transition-like switches of the firing rate with a possible increase of the temperature during the SSL-to-AH bifurcation transition. [ABSTRACT FROM AUTHOR]

Flexible behaviors are organized by complex neural networks in the prefrontal cortex. Recent studies have suggested that such networks exhibit multiple dynamical states, and can switch rapidly from one state to another. In many complex systems such as the brain, the early-warning signals that may predict whether a critical threshold for state transitions is approaching are extremely difficult to detect. We hypothesized that increases in firing irregularity are a crucial measure for predicting state transitions in the underlying neuronal circuits of the prefrontal cortex. We used both experimental and theoretical approaches to test this hypothesis. Experimentally, we analyzed activities of neurons in the prefrontal cortex while monkeys performed a maze task that required them to perform actions to reach a goal. We observed increased firing irregularity before the activity changed to encode goal-to-action information. Theoretically, we constructed theoretical generic neural networks and demonstrated that changes in neuronal gain on functional connectivity resulted in a loss of stability and an altered state of the networks, accompanied by increased firing irregularity. These results suggest that assessing the temporal pattern of neuronal fluctuations provides important clues regarding the state stability of the prefrontal network. We also introduce a novel scheme that the prefrontal cortex functions in a metastable state near the critical point of bifurcation. According to this scheme, firing irregularity in the prefrontal cortex indicates that the system is about to change its state and the flow of information in a flexible manner, which is essential for executive functions. This metastable and/or critical dynamical state of the prefrontal cortex may account for distractibility and loss of flexibility in the prefrontal cortex in major mental illnesses such as schizophrenia. [ABSTRACT FROM AUTHOR]

In this paper, we propose in uncertain dynamical systems a novel method for identifying the region that includes all possible bifurcation boundaries. First, we formulate a robust bifurcation analysis problem for parameter-dependent differential equations with dynamic uncertainties. Next, to solve this problem, we give stability and instability conditions for the uncertain system. Finally, on the basis of these conditions, we propose a method for solving the robust bifurcation analysis problem. [ABSTRACT FROM AUTHOR]

Abstract: The square-wave burster (Wang and Rinzel, 2003) is a class of autonomous bursting cells that share a bifurcation structure. It is known that this class of cells is involved in the generation of various life-supporting rhythms. In our research to realize an electronic circuit that mimics the rhythm generating mechanism in the square-wave burster, our circuit experimentally exhibited severe fluctuations in its rhythmic activity. We have found a noise-sensitive region in the phase portrait of the ideal model and have proposed modifications of the model that can reduce this fluctuation. A possible modification to ionic-conductance neuron models (Kohno and Aihara, 2011) was inspired by them. This modification, however, cannot be applied to a group of square-wave bursters, including the Butera–Rinzel–Smith model (Butera et al., 1999; Del Negro et al., 2001), which is a model of the pre-Bötzinger complex bursting neuron that plays a crucial role in the generation of respiration rhythms, because this modification premises that the slow dynamics originates from an activation gate variable of a hyperpolarizing ionic current. However, in some square-wave bursters, they are controlled by an inactivation gate variable of a depolarizing ionic current. In this study, we proposed a similar modification with a completely different mechanism that can be applied to this group of square-wave bursters. In the presence of noises, the modified Butera–Rinzel–Smith model can generate rhythmic activity that is more stable and similar to biological observations than the original model. The mechanisms underlying this modification are explained with noisy bifurcation diagrams. [Copyright &y& Elsevier]

Inferior olive (IO) neurons project to the cerebellum and contribute to motor control. They can show intriguing spatio-temporal dynamics with rhythmic and synchronized spiking. IO neurons are connected to their neighbors via gap junctions to form an electrically coupled network, and so it is considered that this coupling contributes to the characteristic dynamics of this nucleus. Here, we demonstrate that a gap junction-coupled network composed of simple conductance-based model neurons (a simplified version of a Hodgkin–Huxley type neuron) reproduce important aspects of IO activity. The simplified phenomenological model neuron facilitated the analysis of the single cell and network properties of the IO while still quantitatively reproducing the spiking patterns of complex spike activity observed by simultaneous recording in anesthetized rats. The results imply that both intrinsic bistability of each neuron and gap junction coupling among neurons play key roles in the generation of the spatio-temporal dynamics of IO neurons. [ABSTRACT FROM AUTHOR]

For several decades, androgen suppression has been the principal modality for treatment of advanced prostate cancer. Although the androgen deprivation is initially effective, most patients experience a relapse within several years due to the proliferation of so-called androgen-independent tumor cells. Bruchovsky et al. suggested in animal models that intermittent androgen suppression (IAS) can prolong the time to relapse when compared with continuous androgen suppression (CAS). Therefore, IAS has been expected to enhance clinical efficacy in conjunction with reduction in adverse effects and improvement in quality of life of patients during off-treatment periods. This paper presents a mathematical model that describes the growth of a prostate tumor under IAS therapy based on monitoring of the serum prostate-specific antigen (PSA). By treating the cancer tumor as a mixed assembly of androgen-dependent and androgen-independent cells, we investigate the difference between CAS and IAS with respect to factors affecting an androgen-independent relapse. Numerical and bifurcation analyses show how the tumor growth and the relapse time are influenced by the net growth rate of the androgen-independent cells, a protocol of the IAS therapy, and the mutation rate from androgen-dependent cells to androgen-independent ones. [ABSTRACT FROM AUTHOR]

We analyze a two-dimensional Hindmarsh–Rose type model exhibiting properties of both Class 1 and Class 2 neurons. Although the system is two-dimensional and contains only four parameters, the obtained bifurcation diagrams show that the bifurcation structure satisfies conditions for emergence of both features with constant stimuli. [ABSTRACT FROM AUTHOR]

We propose a two-dimensional neuron model using metal-oxide-semiconductor field-effect-transistors (MOSFETs). It can be implemented with a compact circuitry. This is because we designed its phase portrait structure utilizing the curves native to the MOSFET circuitries. Bifurcation theory is applied to tune the characteristics of the circuitry. Biological neurons can be divided into two main classes based on their responses to sustained current stimuli: classes I and II. At the onset of firing, the firing frequency of class I neurons is asymptotically zero, while that of class II neurons is nonzero. It is generally accepted that classes I and II excitabilities are produced by a saddle-node bifurcation and a subcritical Hopf bifurcation, respectively. Class I* neurons, which belong to a subclass of class I, are characterized by a phase plane structure called a narrow channel. By changing a few circuit parameters, our circuitry can achieve class I without a narrow channel, class I*, and class II. The proposed circuitry is compatible with standard complementary metal-oxide-semiconductor (CMOS) processes. Hence, it can be easily implemented in an analog very-large-scale integrated (VLSI) circuit; further, large networks can be constructed by using this circuitry. We implemented and tested our circuitry, that was constructed using discrete components. We analyzed the responses to singlet, doublet, periodic pulses, and sustained current stimuli; further, we demonstrated that our circuitry inherited three critical properties of biological neurons: (a) the fundamental abilities of excitable cells, (b) neural excitability such as of classes I and II, and (c) an ability to generate chaotic responses to periodic pulse stimuli. We then applied the bifurcation theory to our circuitry and verified its mathematical structure using XPPAUT software. Furthermore, the simulation results on a gap junction (GJ)-coupled network comprising class I* neurons revealed the genesis of itinerant dynamics similar to that found by Fujii et al.. [ABSTRACT FROM AUTHOR]

We propose a method for the design of electronic bursting neurons, based on a simple conductance neuron model. A burster is a particular class of neuron that displays fast spiking regimes alternating with resting periods. Our method is based on the use of an electronic circuit that implements the well-known Morris–Lecar neuron model. We use this circuit as a tool of analysis to explore some regions of the parameter space and to contruct several bifurcation diagrams displaying the basic dynamical features of that system. These bifurcation diagrams provide the initial point for the design and implementation of electronic bursting neurons. By extending the phase space with the introduction of a slow driving current, our method allows to exploit the bistabilities which are present in the Morris–Lecar system to the building of different bursting models. [ABSTRACT FROM AUTHOR]

The present paper studies regular and complex spatiotemporal behaviors in networks of coupled map-based bursting oscillators. In-phase and antiphase synchronization of bursts are studied, explaining their underlying mechanisms in order to determine how network parameters separate them. Conditions for emergent bursting in the coupled system are derived from our analysis. In the region of emergence, patterns of chaotic transitions between synchronization and propagation of bursts are found. We show that they consist of transient standing and rotating waves induced by symmetry-breaking bifurcations, and can be viewed as a manifestation of the phenomenon of chaotic itinerancy. [ABSTRACT FROM AUTHOR]

We have constructed a nerve membrane using MOSFET circuitry, which can be a basic element of an FET-based neural system. Its mechanism of action potentials generation is designed to reproduce that of the Hodgkin-Huxley equations. The responses to singlet, doublet, repetitive pulse, and sustained stimuli are analyzed to show that it exhibits similar properties to the Hodgkin-Huxley equations; namely, 1) excitable dynamics with generation of action potentials, 2) the existence of a chaotic response to periodic stimuli, and 3) Class 2 excitability. It is known that Class. 2 excitability is generated by an inverted Hopf bifurcation. We have applied Hopf bifurcation theory to our nerve membrane's system equations and have shown a routine for ascertaining whether a certain parameter set generates an inverted Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The present paper proposes two types of parametrically coupled circle map networks for multistate associative memory. One of the networks uses a circle map exhibiting an attractor-merging crisis of multiple chaotic attractors to represent a multistate element. The other uses another circle map whose bifurcation diagram serves as a substitute for a multilevel activation function. The configuration of each network is suitably selected according to the dynamics of the individual circle map so that the network can bring about self-organizing chaotic dynamics with an association of a memory. Namely, the coupling term is determined by the generalized partial error function in the first network, and by the weighted sum of inputs in the second network. These multistate networks can be considered as extensions of two kinds of interesting binary networks called the parametrically coupled sine map networks [Lee & Farhat, 2001a], respectively. We illustrate that the proposed networks can exhibit desirable associative dynamics that is missing in the conventional multistate networks. [ABSTRACT FROM AUTHOR]

To clarify the mechanism of the information processing in the brain of living organisms, and investigate information coding of a neuron, a reasonable mathematical model of a neuron is needed. Various models have been proposed and analyzed for now, some relationship between bursting response and bifurcations. In this paper, we propose a simple and effective model to realize bursting phenomena in Amari-Hopfield coupled neurons. In this model, we can constructively design desirable bursting responses. Bifurcation diagrams and some analytical results are given, and a basic design scheme to generate a bursting is shown. © 2003 Wiley Periodicals, Inc. Electr Eng Jpn, 146(2): 43–53, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.10217 [ABSTRACT FROM AUTHOR]

In this paper, we study bifurcation structures of period-adding phenomena in an internal wave model that is a mathematical model for ocean internal waves. It has been suggested that chaotic solutions observed in the internal wave model may be related to the universal property of the energy spectra of ocean internal waves. In numerical bifurcation analyses of the internal wave model, we illustrate bifurcation routes to chaos and parameter regions where chaotic behavior is observed. Furthermore, it is found that the chaotic solutions are related to the period-adding sequence, that is, successive generations of periodic solutions with longer periods as a control parameter is changed. Considering the period-adding sequence as successive local bifurcations, we discuss a mechanism of the phenomena from the viewpoint of bifurcation analysis. We also consider similarity between period-adding phenomena in the internal wave model and ones in the Lorenz model. [ABSTRACT FROM AUTHOR]

We investigate bifurcations in responses of two Hodgkin–Huxley neurons coupled to each other and forced by a periodic pulse train, through the characteristics of synaptic transmissions with an α-function and a time delay. Based on a computational method we previously presented, we show a mechanism of transitions among various kinds of nonperiodic as well as harmonic and subharmonic frequency-locked oscillations. The bifurcation analysis clarifies dependence of sub- and supra-threshold dynamics on coupling and forcing effects. [ABSTRACT FROM AUTHOR]

Presents a study that developed a model for genetic regulatory networks with time delays by functional differential equation. Description of the delay differential equation model; Analysis of the nonlinear properties of the model in terms of local stability and bifurcation; Examination of the stability of the equilibrium state of the network with time delays.

We investigate a noninvertible map describing burst firing in a chaotic neural network model with ring structure. Since each neuron interacts with many other neurons in biological neural systems, it is important to consider global dynamics of networks composed of nonlinear neurons in order to clarify not only mechanisms of emergence of the burst firing but also its possible functional roles. We analyze parameter regions in which burst firing can be observed, and show that dynamics of strange attractors with burst firing is related to the generation of a homoclinic-like situation and vanishing of an invariant closed curve of the map. [ABSTRACT FROM AUTHOR]

The agricultural tractor-vibrating subsoiler system behaves as an impact oscillator. The bifurcation structure of vibrations of the system are investigated by field experiments by varying the forcing frequency and observing the periodic, period-doubling and chaotic vibrations. The position-triggered Poincaré sections are obtained by generating signals for a sampling timing and the structure of a strange attractor is clearly demonstrated on the Poincaré sections. The period-doubling route to chaos is identified experimentally for the tested tractor-implement system. [ABSTRACT FROM AUTHOR]

This paper presents a global bifurcation scenario for chaotic dynamical systems that solve optimization problems. First, a chaotic dynamical system is constructed by a discretization of the gradient-descent dynamical system of the objective function. With an increase in the discretization parameter, local minimum solutions of the objective function bifurcate into chaotic attractors through a period-doubling bifurcation. The chaotic attractors are initially localized in the state space and eventually merge into a single global chaotic attractor via a series of crises. On the global chaotic attractor which visits a variety of local minima, “chaotic search” for a global minimum is realized. On the basis of the bifurcation scenario, we provide a guideline for tuning the bifurcation parameter value which gives rise to an efficient “chaotic search.” We also consider the efficiency of the “chaotic simulated annealing” algorithm in the light of its annealing schedule. © 1998 Scripta Technica. Electron Comm Jpn Pt 3, 81(2), 1–12, 1998 [ABSTRACT FROM AUTHOR]

A degenerate optical parametric oscillator (DOPO) network realized as a coherent Ising machine can be used to solve combinatorial optimization problems. Both theoretical and experimental investigations into the performance of DOPO networks have been presented previously. However a problem remains, namely that the dynamics of the DOPO network itself can lower the search success rates of globally optimal solutions for Ising problems. This paper shows that the problem is caused by pitchfork bifurcations due to the symmetry structure of coupled DOPOs. Some two-parameter bifurcation diagrams of equilibrium points express the performance deterioration. It is shown that the emergence of non-ground states regarding local minima hampers the system from reaching the ground states corresponding to the global minimum. We then describe a parametric strategy for leading a system to the ground state by actively utilizing the bifurcation phenomena. By adjusting the parameters to break particular symmetry, we find appropriate parameter sets that allow the coherent Ising machine to obtain the globally optimal solution alone. [ABSTRACT FROM AUTHOR]

Glacial–interglacial cycles are large variations in continental ice mass and greenhouse gases, which have dominated climate variability over the Quaternary. The dominant periodicity of the cycles is ∼ 40 kyr before the so-called middle Pleistocene transition between ∼ 1.2 and ∼ 0.7 Myr ago, and it is ∼ 100 kyr after the transition. In this paper, the dynamics of glacial–interglacial cycles are investigated using a phase oscillator model forced by the time-varying incoming solar radiation (insolation). We analyze the bifurcations of the system and show that strange nonchaotic attractors appear through nonsmooth saddle–node bifurcations of tori. The bifurcation analysis indicates that mode-locking is likely to occur for the 41 kyr glacial cycles but not likely for the 100 kyr glacial cycles. The sequence of mode-locked 41 kyr cycles is robust to small parameter changes. However, the sequence of 100 kyr glacial cycles can be sensitive to parameter changes when the system has a strange nonchaotic attractor. [ABSTRACT FROM AUTHOR]

Abstract: We analyse a piecewise-linear oscillator that consists of a three- resonant circuit with a hysteresis element. Three sets of two-dimensional linear equations, including a hysteresis function, represent the governing equations of the circuit, and all the Lyapunov exponents are calculated in a remarkably simple manner based on derived explicit solutions. Various dynamical phenomena such as two-torus, three-torus, and hyperchaos with four positive Lyapunov exponents are observed by Lyapunov analysis. We obtained a detailed bifurcation diagram in which novel bifurcation structure which we call a “two-torus Arnold tongue” is observed where two-torus generating regions exist in a three-torus generating region as if periodic states exist in a two-torus generating region. [Copyright &y& Elsevier]

Abstract: Torus-doubling bifurcations typically occur only a finite number of times. It has been assumed that torus-doubling bifurcations in quasiperiodically forced systems are interrupted by the appearance of strange nonchaotic attractors (SNAs). In the present Letter, we study a quasiperiodically forced noninvertible map and report the occurrence of a torus-doubling process via SNAs. The mechanism of this process is numerically clarified. Furthermore, this process is experimentally demonstrated in a switched-capacitor integrated circuit. [Copyright &y& Elsevier]

In this paper, we analyze the sudden change from chaos to oscillation death generated by the Bonhoeffer-van der Pol (BVP) oscillator under weak periodic perturbation. The parameter values of the BVP oscillator are chosen such that a stable focus and a stable relaxation oscillation coexist if no perturbation is applied. In such a system, complicated bifurcation structure is expected to emerge when weak periodic perturbation is applied because the stable focus and the stable relaxation oscillation coexist in close proximity in the phase plane. We draw a bifurcation diagram of the fundamental harmonic entrainment. The bifurcation structure is complex because there coexist two bifurcation sets. One is the bifurcation set generated in the vicinity of the stable focus, and the other is that generated in the vicinity of the stable relaxation oscillation. By analyzing the bifurcation diagram in detail, we can explain the sudden change from chaos with complicated waveforms to oscillation death. We make it clear that this phenomenon is caused by a saddle-node bifurcation. [ABSTRACT FROM AUTHOR]

Abstract: Hybrid systems are widely used to model dynamical phenomena that are characterized by interplay between continuous dynamics and discrete events. An example of biomedical application is modeling of disease progression of prostate cancer under intermittent hormonal therapy, where continuous tumor dynamics is switched by interruption and reinstitution of medication. In the present paper, we study a hybrid systems model representing intermittent androgen suppression (IAS) therapy for advanced prostate cancer. Intermittent medication with switching between on-treatment and off-treatment periods is intended to possibly prevent a prostatic tumor from developing into a hormone-refractory state and is anticipated as a possible strategy for delaying or hopefully averting a cancer relapse which most patients undergo as a result of long-term hormonal suppression. Clinical efficacy of IAS therapy for prostate cancer is still under investigation but at least worth considering in terms of reduction of side effects and economic costs during off-treatment periods. In the model of IAS therapy, it depends on some clinically controllable parameters whether a relapse of prostate cancer occurs or not. Therefore, we examine nonlinear dynamics and bifurcation structure of the model by exploiting a numerical method to clarify bifurcation sets in the hybrid system. Our results suggest that adjustment of the normal androgen level in combination with appropriate medication scheduling could enhance the possibility of relapse prevention. Moreover, a two-dimensional piecewise-linear system reduced from the original model highlights the origin of nonlinear phenomena specific to the hybrid system. [Copyright &y& Elsevier]

Abstract: Circadian oscillations with a period of about 24h are observed in nearly all living organisms as conspicuous biological rhythms. In this paper, we investigate various kinds of bifurcation phenomena produced in a circadian oscillator model of Drosophila. In Drosophila, it is known that circadian oscillations in the levels of two proteins, PER and TIM, result from the negative feedback exerted by a PER–TIM complex on the expression of the per and tim genes that code for the two proteins. For studying circadian oscillations of proteins in Drosophila, a mathematical model has been proposed. The model cannot only account for regular circadian oscillations in environmental conditions such as constant darkness, but also give rise to more complex oscillatory phenomena including chaos and birhythmicity. By calculating bifurcations using Kawakami''s method, we obtain detailed bifurcation diagrams related to stable and unstable invariant sets, and identify parameter regions in which the model generates complex oscillations as well as regular circadian oscillations. Moreover, we study bifurcations observed in the model incorporating the effect on a light–dark (LD) cycle and show that the waveform of the periodic variation in the light-induced parameter has a marked influence on the global bifurcation structure or the type of dynamic behavior resulting from the forcing term of the circadian oscillator by the LD cycles. [Copyright &y& Elsevier]

Abstract: The Morris–Lecar (M–L) equations are an important neuron model that exhibits classes I and II excitabilities when system parameters are set appropriately. Although many papers have clarified characteristic behaviors of the model, the detailed transition between two classes is unclear from the viewpoint of bifurcation analyses. In this paper, we investigate bifurcations of invariant sets in a five-dimensional parameter space, and identify an essential parameter of the half-activated potential of the potassium activation curve that contributes to the alternation of the membrane properties of the M–L neuron. We also show that the membrane property can be controlled by varying the value of the single parameter. [Copyright &y& Elsevier]