Stochastic global bifurcation in perturbed Hamiltonian systems
Lora Billings Erik M. Bollt David Morgan Ira B. Schwartz
Conference Publications 2003, 2003(Special): 123-132 doi: 10.3934/proc.2003.2003.123
We study two perturbed Hamiltonian systems in which chaos-like dynamics can be induced by stochastic perturbations. We show the similarities of a class of population and laser models, analytically and topologically. Both systems have similar manifold structure that includes bi-instability and partially formed heteroclinic connections. Noise takes advantage of this structure, inducing a global bifurcation and chaotic-like dynamics which exhibits mixed mode behavior of the original bi-stable solutions. We support these claims with numerical approximations of the transport between basins.
keywords: Stochastic dynamical systems chaos. bifurcation Hamiltonian
Detecting phase transitions in collective behavior using manifold's curvature
Kelum Gajamannage Erik M. Bollt
Mathematical Biosciences & Engineering 2017, 14(2): 437-453 doi: 10.3934/mbe.2017027

If a given behavior of a multi-agent system restricts the phase variable to an invariant manifold, then we define a phase transition as a change of physical characteristics such as speed, coordination, and structure. We define such a phase transition as splitting an underlying manifold into two sub-manifolds with distinct dimensionalities around the singularity where the phase transition physically exists. Here, we propose a method of detecting phase transitions and splitting the manifold into phase transitions free sub-manifolds. Therein, we firstly utilize a relationship between curvature and singular value ratio of points sampled in a curve, and then extend the assertion into higher-dimensions using the shape operator. Secondly, we attest that the same phase transition can also be approximated by singular value ratios computed locally over the data in a neighborhood on the manifold. We validate the Phase Transition Detection (PTD) method using one particle simulation and three real world examples.

keywords: Phase transition manifold collective behavior dimensionality reduction curvature
Communication and Synchronization in Disconnected Networks with Dynamic Topology: Moving Neighborhood Networks
Joseph D. Skufca Erik M. Bollt
Mathematical Biosciences & Engineering 2004, 1(2): 347-359 doi: 10.3934/mbe.2004.1.347
We consider systems that are well modelled as networks that evolve in time, which we call Moving Neighborhood Networks. These models are relevant in studying cooperative behavior of swarms and other phenomena where emergent interactions arise from ad hoc networks. In a natural way, the time-averaged degree distribution gives rise to a scale-free network. Simulations show that although the network may have many noncommunicating components, the recent weighted time-averaged communication is sufficient to yield robust synchronization of chaotic oscillators. In particular, we contend that such time-varying networks are important to model in the situation where each agent carries a pathogen (such as a disease) in which the pathogen's life-cycle has a natural time-scale which competes with the time-scale of movement of the agents, and thus with the networks communication channels.
keywords: communication in complex networks. mathematical model Disease spread in communities nonlinear dynamics self-organization epidemiology
Control entropy: A complexity measure for nonstationary signals
Erik M. Bollt Joseph D. Skufca Stephen J . McGregor
Mathematical Biosciences & Engineering 2009, 6(1): 1-25 doi: 10.3934/mbe.2009.6.1
We propose an entropy statistic designed to assess the behavior of slowly varying parameters of real systems. Based on correlation entropy, the method uses symbol dynamics and analysis of increments to achieve sufficient recurrence in a short time series to enable entropy measurements on small data sets. We analyze entropy along a moving window of a time series, the entropy statistic tracking the behavior of slow variables of the data series. We employ the technique against several physiological time series to illustrate its utility in characterizing the constraints on a physiological time series. We propose that changes in the entropy of measured physiological signal (e.g. power output) during dynamic exercise will indicate changes in underlying constraint of the system of interest. This is compelling because CE may serve as a non-invasive, objective means of determining physiological stress under non-steady state conditions such as competition or acute clinical pathologies. If so, CE could serve as a valuable tool for dynamically monitoring health status in a wide range of non-stationary systems.
keywords: Entropy signal analysis Physiology
Heart rate variability as determinism with jump stochastic parameters
Jiongxuan Zheng Joseph D. Skufca Erik M. Bollt
Mathematical Biosciences & Engineering 2013, 10(4): 1253-1264 doi: 10.3934/mbe.2013.10.1253
We use measured heart rate information (RR intervals) to develop a one-dimensional nonlinear map that describes short term deterministic behavior in the data. Our study suggests that there is a stochastic parameter with persistence which causes the heart rate and rhythm system to wander about a bifurcation point. We propose a modified circle map with a jump process noise term as a model which can qualitatively capture such this behavior of low dimensional transient determinism with occasional (stochastically defined) jumps from one deterministic system to another within a one parameter family of deterministic systems.
keywords: circle map next angle map electrocardiography jump process. Heart rate variability
Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies
Rakesh Pilkar Erik M. Bollt Charles Robinson
Mathematical Biosciences & Engineering 2011, 8(4): 1085-1097 doi: 10.3934/mbe.2011.8.1085
Bursts of 2.5mm horizontal sinusoidal anterior-posterior oscillations of sequentially varying frequencies (0.25 to 1.25 Hz) are applied to the base of support to study postural control. The Empirical Mode Decomposition (EMD) algorithm decomposes the Center of Pressure (CoP) data (5 young, 4 mature adults) into Intrinsic Mode Functions (IMFs). Hilbert transforms are applied to produce each IMF’s time-frequency spectrum. The most dominant mode in total energy indicates a sway ramble with a frequency content below 0.1 Hz. Other modes illustrate that the stimulus frequencies produce a ‘locked-in’ behavior of CoP with platform position signal. The combined Hilbert Spectrum of these modes shows that this phase-lock behavior of APCoP is more apparent for 0.5, 0.625, 0.75 and 1 Hz perturbation intervals. The instantaneous energy profiles of the modes depict significant energy changes during the stimulus intervals in case of lock-in. The EMD technique provides the means to visualize the multiple oscillatory modes present in the APCoP signal with their time scale dependent on the signals’s successive extrema. As a result, the extracted oscillatory modes clearly show the time instances when the subject’s APCoP clearly synchronizes with the provided sinusoidal platform stimulus and when it does not.
keywords: Center of Pressure Empirical Mode Decomposition Induced Oscillations. Posture and Balance Sinusoidal Per- turbations

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